IITJEE – MAINS & ADVANCED AIIMS / AIPMT Concept of Centre of Mass - Physics...

IITJEE – MAINS & ADVANCED AIIMS / AIPMT

Concept of

Centre of Mass

Q12. The C.M. of a system of particles is at the origin. It follows that:

(a) The number of particles to the right of the origin is equal to the number of particles to the left. (b) The total mass of the particles to the right of the origin is same as the total mass of the particles to the left of the origin. (c) The number of particles on X-axis should be equal to the number of particles on Y-axis (d) If there is a particle on the positive X-axis ; there must be at least one particle on the negative X-axis

Q13. A body has its C.M. at the origin. The X-coordinates of the particles;

(a) May all be positive (b) May be all negative (c) May be all non-negative (d) May be positive for some cases and negative for in other cases.

Q38. Four point masses A, B and C are placed at points which are coplanar but non-collinear. If P is the CM of the system comprising A, B, C and D, then

(a) P must lie within the quadrilateral ABCD (b) P must lie on the line joining any two points A, B, C and D. (c) P may or may not coincide with one of the point masses. (d) P lies inside or on the edge of at least one of the triangles formed by taking any three of the points from A, B, C and D.

Q42. A system consists of two point masses M and m (< M). The centre of mass of the system is:

a)  At the middle of the of m and M b)  Closer to m c)  Closer to M d)  At the position of larger mass.

Q14. In which of the following cases the C.M. of a rod is certainly not at its centre?

(a) The density continuously increases from left to right. (b) The density continuously decreases from left to right. (c) The density decreases from left to right upto the centre and then increases. (d) The density increases from left to right upto the centre and then decreases.

Q1. The position vector of three particles of masses m1 = 1 kg, m2 = 2 kg and m3 = 3 kg are r1 = î + 4ĵ + ǩ, r2 = î + ĵ + ǩ and r3 = 2î – ĵ – 2ǩ respectively. Find the position vector of their center of mass.

Q2. Three particles of masses 0.5kg, 1kg and 1.5 kg are placed at the three corners of a right angled triangle at sides 3cm, 4cm and 5cm as shown. Locate the C.M. of the system.

Q4 A cubical block of ice of mass m and edge length L is placed in a large tray of mass M. If the ice melts, how far does the C.M. of the system “ice + tray” come down?

Q22. Consider a gravity free hall in which a tray of mass M, carrying a cubical block of ice of mass m and edge L, is at rest in the middle. If the ice melts, by what distance does the C.M. of the “tray + ice” system descend?

Q10. A circular plate of diameter d is kept in contact with a square plate of edge d as shown. The density of material and the thickness are same everywhere. The centre of mass the composite system will be:

(a) Inside the circular plate (b) Inside the square plate (c) At the point of contact (d) Outside the system.

Q18. A square plate of edge d and a circular disc of d iameter d are p laced touching each other as shown. Locate the C.M. of the combination, assuming same mass per unit area of the two plates.

Q3. Half of the rectangular plate shown in the figure is made up of a material of density ρ1 and other of density ρ2. The length of the plate is L. Locate the C.M. of plate.

Q16. The structure of a water molecule is shown in figure. Find the distance of C.M. of the molecule from the centre of oxygen atom.

Q39. Three point masses 1g, 2g and 3g have their center of mass at (2, 2, 2) respectively. A fourth mass of 4g is placed at such a point that the new CM is at (0, 0, 0). The coordinate of the 4th mass is

(a) (– 1, – 1, – 1) (b) (– 2, – 2, – 2) (c) (– 3, – 3, – 3) (d) (– 4, – 4, – 4)

Q17. Seven homogeneous bricks, each of length L, are arranged as shown in figure. Each brick is displaced wrt one in contact by L/10. Find the co-ordinate of the C.M. relative to the origin as shown.

Q20. Two blocks of masses 10kg and 20kg are placed on the X-axis. The first mass is moved on the axis by a distance of 2cm. By what distance should the second mass be moved to keep the position of C.M. unchanged?

Q21. Two blocks of mass 10kg and 30kg are placed along the straight line. The first block is raised through a height of 7cm. By what distance should the second mass be moved to raise the C.M. by 1cm?

Q9. All the particles are situated at a distance R from the origin. The distance of the centre of mass from the origin is

(a) = R (b) ≤ R (c) ≥ R (d) > R.

Q26. Three masses 2kg, 3kg and 4kg are lying on the corners o f an equ i la t e ra l triangle of side a as s h o w n . T h e x -coordinate of CM is at

Q. A thin uniform wire is bent to form the two equal sides AB and AC of triangle ABC, where AB = AC = 5 cm. The third side BC, of length 6cm, is made from uniform wire of twice the density of the first. The distance of centre of mass from A is:

a)  34/11 cm * b)  34/9 cm c)  11/34 cm d)  11/45 cm

Q. Five homogeneous bricks, each of length L, are arranged as shown in figure. Each brick is displaced with respect to the one in contact by L/5. Find the x-coordinate of the centre of mass relative to the origin O shown.

RESONANCE CENTRE OF MASS - 3

A 4. Five homogeneous bricks, each of length L, are arranged as shown in figure. Each brick is displacedwith respect to the one in contact by L/5. Find the x-coordinate of the centre of mass relative to theorigin O shown.i k¡p l eku bZaVas çR; sd dh yEckbZ L,dks fp=kkuql kj O; ofLFkr fd; k x; k gSA çR; sd bZaV dks bl ds l Ei dZ okyh bZaV ds l ki s{kL/5 l s foLFkkfi r fd; k x; k gSA fcUnq O ds l ki s{k nzO; eku dsUnz dk x-funsZ’kkad Kkr dfj ; sA

Ans. 50L33

Sol.

COM of brick 1 and 5 → 2L

1 o 5 bZV dk nzO; eku dsUnz → 2L

COM of brick 2 and 4 → 2L

+ 5L

2 o 4 bZV dk nzO; eku dsUnz → 2L

+ 5L

COM of brick 3 → 2L

+ 5L2

3 bZV dk nzO; eku dsUnz → 2L

+ 5L2

Xcm = m5

5L2

2Lm

5L

2Lm2

2Lm2 ⎟

⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ ++

= 50L33

A 5. A uniform disc of radius R is put over another uniform disc of radius 2R made of same materialhaving same thickness. The peripheries of the two discs touches each other. Locate the centre ofmass of the system taking center of large disc at origin.R f=kT; k dh l e: i pdrh dks l eku i nkFkZ rFkk l eku eksVkbZ dh 2R f=kT; k okyh nwl jh l e: i pdrh i j j [krs gSAnksuksa pdfr; ksa dh i fj f/k , d nwl js dks Li ’kZ djrh gSA cM+h pdrh ds dsUnz dks ewy fcUnq ekurs gq; s fudk; ds nzO; ekudsUnz dh fLFkfr crkb; sAAns. At R/5 from the centre of the bigger disc towards the centre of the smaller disk.cM+h pdrh ds dsUnz l s R/5 nwjh i j NksVh pdrh ds dsUnz dh r jQA

Sol.

M1 = π (2R)2 × ρ M2 = π (R)2 × ρ x1 = 0, x2 = R

So bl fy, Xcm = 21

2211mm

xmxm++

Xcm = ρ×π+ρ×π

×ρπ+×ρπ22

22

RR4RR0R4

= 5R

towards smaller disc

NksVh pdrh dh vkSj

Q. A uniform solid cone of height 40 cm is shown in figure. The distance of centre of mass of the cone from point B (centre of the base) is:

a)  20 cm b)  10/3 cm c)  20/3 cm d)  10 cm *


A-5. A uniform solid cone of height 40 cm is shown in figure. The distance of centre of mass of the conefrom point B (centre of the base) is :, d l eku Bksl ’kadq ft l dh ÅpkbZ 40 l seh- gS] fp=kkuql kj fn[ kk; k x; k gSA fcUnq B l s ’kadw ds nzO; eku dsUnz dhnwjh ¼vk/kkj ds dsUnz l s½ gksxh &

(A) 20 cm (B) 10/3 cm (C) 20/3 cm (D*) 10 cm

A-6. The centre of mass of a system of particles is at the origin. From this we conclude that(A) The number of particles on positive x-axis is equal to the number of particles on negative x-axis(B) The total mass of the particles on positive x-axis is same as the total mass on negative x-axis(C*) The number of particles on X-axis may be equal to the number of particles on Y-axis.(D) If there is a particle on the positive X-axis, there must be at least one particle on the negative X-axis.

fdl h d.k&fudk; dk nzO; eku dsUnz ewy fcUnq i j gSA bl dk vfHkçk; gS] fd &(A) /kukRed x–v{k i j d.kksa dh l a[ ; k] _ .kkRed x–v{k i j d.kksa dh l a[ ; k ds cjkcj gksxhA(B) /kukRed x–v{k i j d.kksa dk dqy nzO; eku] _ .kkRed x–v{k i j d.kksa ds dqy nzO; eku ds cjkcj gksxkA(C*) x–v{k i j d.kksa dh dqy l a[ ; k] Y–v{k i j d.kksa dh dqy l a[ ; k ds cjkcj gks l drh gSA(D) ; fn ?kukRed x–v{k i j dksbZ d.k gksxk rks de l s de , d d.k _ .kkRed x–v{k i j vo’ ; gksxkA

A-7.* A system of particles has its centre of mass at the origin. The x-coordinates of all the particles(A) may be positive(B) may be negative(C*) may be non-negative(D*) may be non-positive, d d.k fudk; dk nzO; eku dsUnz ewy&fcUnq i j gSA l Hkh d.kksa ds x–funsZ’kkad &(A) /kukRed gks l drs gSA (B) _ .kkRed gks l drs gSA(C*) v&_ .kkRed gks l drs gSA (D*) v&/kukRed gks l drs gSA

A-8.* In which of the following cases the centre of mass of a system is certainly not at its centre ?(A*) A rod whose density continuously increases from left to right(B*) A rod whose density continuously decreases from left to right(C) A rod whose density decreases from left to right upto the centre and then increases(D) A rod whose density increases from left to right upto the centre and then decreases

fuEu es l s fdu fLFkfr; ksa ds fy; s fudk; dk nzO; eku dsUnz fuf’pr : i l s bl ds dsUnz i j ugha gksxk &(A) , d NM ft l dk ?kuRo cka; h l s nka; h vksj fujar j c<+rk jgsA(B) , d NM ft l dk ?kuRo cka; h l s nka; h vksj fujUraj de gksrk jgsaA(C) , d NM ft l dk ?kuRo cka; h l s nka; h vksj e/; fcUnq rd de gksrk jgs rRi ’pkr~ c<+sA(D) , d NM ft l dk ?kuRo cka; h l s nka; h vksj e/; fcUnq rd c<+rk jgs rRi ’pkr~ de gksrk jgsA

Q29. A thin rod of length L and mass m has a disc which is attached to one of its ends such that the rod and the disc are coplanar. Mass of the disc is M and radius is r. CM from the centre of disc is ____________.

Q36. Masses of 1g, 2g, 3g, ……, 100g are suspended from the 1cm, 2cm, 3cm, ……, 100cm marks of a light meter scale. The system will be supported in equilibrium at

(a) 60cm (b) 66 cm (c) 55 cm (d) 72 cm

Q60. Three identical spheres are kept as shown. If their centres are marked as P, Q and R respectively, then the distance of CM from P is

a)  (PQ + PR + QR)/3 b)  (PQ + QR)/3 c)  (PQ + PR)/3 d)  (PR + QR)/3

Ideal Coaching Centre & Fusion Classes Chapter # 7 Rotational Motion AIEEE/PMT

Prepared'By.:'Er.'Veenus'Girdhar' 4"

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Q45. Four particles of masses m1 = 2kg, m2 = 4kg, m3 =1kg and m4 are placed at four corners of a square as shown in figure. C a n m a s s o f m 4 b e adjusted in such a way that the centre of mass of system will be at the centre of the square, C.

Centre of Mass Web: http://www.locuseducation.org

P H Y S I C S 1 8L O C U S

7. Three laminar objects of same density a square, a disc andan equilateral triangle are placed as shown in figure. Findthe coordinates of the centre of mass of the system of thesethree bodies.

Y

X

lO

l

ll

8. Four particles of masses m1 = 2 kg, m2 = 4 kg, m3 = 1 kg andm4 are placed at four corners of a square as shown in figure.Can mass of m4 be adjusted in such a way that the centre ofmass of system will be at the centre of the square, C. C

m1 m2

m4 m3

9. From a uniform disc of radius R, a circular hole of radius R/2 is cut. The centre of the hole is at R/2 from thecentre of the original disc. Locate the centre of mass of the resulting flat body.

10. From a uniform disc of radius R, two circular sections each of radius R/4 have been removed as shown infigure. For the reference axes shown the co-ordinates of centre of mass of remaining body are

(a)3 3,

112 112R R− −

(b)3 3,56 56

R R− −

(c)3 3,

128 128R R− −

(d)3 3,64 64

R R− −.

x

y

11. Figure shows a thin uniform disc of radius R, from which ahole of radius R/2 has been cut out from left of the centreand is placed on right of the centre of disc. Find the C.M. ofthe resulting system.

(- /2 0)R ,

y

x( /2 0)+R ,

RO

12. A square hole is punched out from a circular lamina, the diagonal of the square being a radius of the circle.Show that the centre of mass of the remaining body is at a distance R/(4π – 2) from the centre of the circle,where R is the radius of the circular lamina.

13. A nonuniform thin rod of length L lies along the x axis with one end at the origin. It has a linear mass densityλ kg/m, given by 0(1 / ).x Lλ λ= + The density is thus twice as great at one end as at the other. (a) Use M

= dm∫ to find the total mass. (b) Find the centre of mass of the rod.

14. AB is a uniformly shaped thin rod of length L, but its linear mass density varies with distance from one end Aof the rod as 2 ,px cλ = + where p and c are positive constants. Find out the distance of the centre of mass ofthis rod from the end A.

15. Use integration to find the centre of mass of the rightisosceles triangle shown in figure.

y

x10 m

10 m

0

Q32. A uniform square plate has a mass of 2 kg. Two masses of 1 kg each are placed at the corners B and C. The C.M. of the system gets shifted to: (See figure)

(a) Mid point of OC (b) Mid point of OB (c) Mid point of OY (d) Mid point of OX

Y

D C

OX

BA

Q33. Masses of 1 kg each are placed at the corners B and C of a uniform square plate ABCD. A mass of 2 kg is to be placed on the plate so that C.M. of the system remains at O. This mass should be placed at (See figure)

(a) P (b) R (c) S (d) Mid point of S and Q.

P

DC

O Q

BA

R

S

Q54. Look at the drawing given in the figure, which has been drawn w i th ink o f un i fo r m l ine -thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is m. The mass of the ink used to draw the outer circle is 6m. The co-ordinates of the centre of different parts are: outer circle (0, 0), left inner circle (– a, a), right inner circle (a, a), vertical line (0, 0) and horizontal line (0, – a). The y co-coordinate of the centre of mass of the ink in this drawing is ________



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Q40. The object shown in figure is constructed using uniform rods of same material. Find the position of C.M. in each case

Q. Calculate entre of mass of the system. Each cube is of mass M and side L.

Q32. A tin sheet of uniform mass distribution is shown in figure. The CM of the sheet, from top left end of the system, is at:

(a) (5, – 0.5) (b) (– 0.5, 5) (c) (10, – 1) (d) (– 1, 10)

Q51. A T shaped object with dimensions shown in the figure, is lying on a smooth floor. A force F is applied at the point P parallel to AB, such that the object has only the translat ional mot ion without rotation. Find the location of P with respect to C.



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Q28. A cone and a sphere is as shown in the figure. Density of material of cone is 1/12 times the density of sphere. The position of C M , f r o m b a s e o f system, on the line of symmetry is _________

Q34. A hemisphere and a solid cone have a common base. The CM of the common structure coincides with the centre of the common base. If R is the radius of hemisphere and h is height of cone, then _________

Q44. Three laminar o b j e c t s o f s a m e density a square, a d i s c a n d a n equilateral triangle are placed as shown in figure. Find the coordinates of the centre of mass of the system of these three bodies.




Y

X

lO

l

ll


m1 m2

m4 m3



(a)3 3,

112 112R R− −

(b)3 3,56 56

R R− −

(c)3 3,

128 128R R− −

(d)3 3,64 64

R R− −.

x

y


(- /2 0)R ,

y

x( /2 0)+R ,

RO






y

x10 m

10 m

0

Q25. A 3kg body has a velocity of (î + 6ĵ) m/s and a 2kg body has a velocity of (2î – ĵ) m/s. Then the velocity of CM in m/s is ____________

Q19. Calculate the velocity of C.M. of the system as shown in figure.

Q. Two masses M and m are hung with the help of a thread passing over a pulley, Find the acceleration of centre fo mass, if M > m.

Q. Two particles having mass ratio n: 1 are interconnected by a light inextensible string that passes over a smooth pulley. I f the system is released, then the acceleration of the centre of mass of the system is:


SECTION (B) : MOTION OF CENTRE OF MASS nzO; eku dsUnz dh xfrB-1. Two particles of mass 1 kg and 0.5 kg are moving in the same direction with speed of 2m/s and 6m/s

respectively on a smooth horizontal surface. The speed of centre of mass of the system is :1 kg rFkk 0.5 kg ds nks nzO; eku , d gh fn’kk esa Øe’k% 2 m/sec rFkk 6 m/sec l s fpdus l rg i j xfr dj jgsgSA fudk; ds nzO; eku dsUnz dh pky gksxhA

(A*) 3

10 m/s (B)

710

m/s (C) 211

m/s (D) 3

12 m/s

Sol. vcm = sec/m3

102/11

621

21=

+

×+×.

B-2. Two particles of equal mass have initial velocities i2 ms–1 and j2 ms–1 . First particle has a constant

acceleration )ji( + ms–2 while the acceleration of the second particle is always zero. The centre ofmass of the two particles moves inl eku nzO; eku ds nks d.k ds çkj fEHkd osx Øe’k% i2 ms–1 o j2 ms–1 gSA i zFke d.k dk fu; r Roj .k )ji( +ms–2 gS t cfd nwl js dk Roj .k l nSo ’kwU; gSA nksuks d.kksa dk nzO; eku dsUnz fuEu oØ i j xfr djsxkA(A) Circle o‘Ùk (B) Parabola i joy;(C) Ellipse nh?kZo‘Ùk (D*) Straight line l jy js[ kk

Sol. vcm = 21

2211mm

vmvm++

∴ vcm = m2

)j2(m)i2(m +

acm = m2

)0(m)ji(m ++.

vcm has same direction as of acmVcm dh fn’kk acm dh fn’kk esa gksxh∴ straight line. l h/kh js[ kk

B-3. Two particles having mass ratio n : 1 are interconnected by a light inextensible string that passes overa smooth pulley. If the system is released, then the acceleration of the centre of mass of the systemis :nks d.k ft uds nzO; eku dk vuqi kr n : 1 gS] os , d gYdh vforkU; jLl h l s t qM+s gq, gSa t ks fpduh f?kjuh l sxqt j rh gSA vxj fudk; dks NksM+ fn; k t krk gS rks l ewg ds nzO; eku dsUnz dk Roj .k gksxkA

(A) (n – 1)2 g (B) g1n1n 2

⎟⎠⎞

⎜⎝⎛

−+

(C*) g1n1n 2

⎟⎠⎞

⎜⎝⎛

+−

(D) g1n1n⎟⎠⎞

⎜⎝⎛

−+

Sol. a = mnm

)m–nm(+

g

= )1n()1–n(

+ g

a1 = a2 = a

acm = )mnm(ma–nma 21

+ = a)1n()1–n(×

+

acm = g)1n(

)1–n(2

2

+.

Q41. A rod of length L is placed along the x-axis between x = 0 and x = L. The linear density (mass/length) λ of the rod varies with the distance x from the origin as λ = Rx. Here, R is a positive constant. Find the position of center of mass of this rod. [2L/3]

Q49. A non-uniform thin rod of length L lies along the x axis with one end at the origin. It has a linear mass density λ kg/m, given by λ = λ0(1 + x/L). The density is thus twice as great at one end as at the other. (a) Find the total mass. (b) Find the centre of mass of the rod.

Q50. AB is a uniformly shaped thin rod of length L, but its linear mass density varies with distance from one end A of the rod as λ = px2 + c, where p and c are positive constants. Find out the distance of the centre of mass of this rod from the end A.

Q61. If the linear density of a rod of length 3m varies as λ = 2 + x, then the position of CG of rod is given by

a)  7/3 cm b)  12/7 cm c)  10/7 cm d)  9/7 cm

Q. The centre of mass of the shaded portion of the disc is:

a)  R/20 to the left of A. * b)  R/12 to the left of A. c)  R/20 to the right of A. d)  R/12 to the right of A.


G 2. Sand drops from a stationary hopper at the rate of 5 kg/s falling on a conveyor belt moving with a constantspeed of 2 m/s. What is the force required to keep the belt moving and what is the power delivered by themotor moving the belt?, d fLFkj gqi j l s , d duos; j csYV t ks 2 m/s dh fu; r pky l s py jgh gS i j 5 kg/s dh nj l s jsr fxjk; h t krh gSAcsYV dks pyk, s j [kus ds fy, fdrus cy dh vko’ ; drk gS vkSj eksVj }kjk csYV dks nh x; h ’kfDr fdruh gS \

Ans. Fext = 10N; P = 20 watt.

Sol. F = µ dtdm

= 2×5 = 10 N Ans.

P = F.v. = 10×2 = 20 W Ans.

PART - II : OBJECTIVE QUESTIONSSECTION (A) : CALCULATION OF CENTRE OF MASS nzO; eku dsUnz dh x.kukA-1. The centre of mass of a body :

oLrq dk nzO; eku dsUnz :(A) Lies always at the geometrical centre (ges’kk T; kferh dsUnz i j fLFkr gksrk gSaA)(B) Lies always inside the body (ges’kk oLrq ds vUnj gksrk gS)(C) Lies always outside the body (ges’kk oLrq ds ckgj gksrk gS)(D*) Lies within or outside the body (oLrq ds vUnj ; k ckgj gksrk gSA)

A-2. The centre of mass of the shaded portion of the disc is : (The mass is uniformly distributed in theshaded portion) :pdrh ds Nk; kafdr gq; s Hkkx dk nzO; eku dsUnz gksxk % (Nk; kafdr Hkkx esa nzO; eku l e: i for fj r gS)

(A*) 20R

to the left of AA (A l s ck; ha vksj 20R i j )

(B) 12R

to the left of A ( AA l s ck; ha vksj 12R i j )

(C) 20R

to the right of AA (A l s nk; ha vksj 20R

i j )

(D) 12R

to the right of A (A l s nk; ha vksj 12R i j )

Sol. A1 = πR2 A2 = 16R2π

x1 = 0 x2 = 4R3

xcen = 20R

–

16R–R

4R3

16R

–0

22

2

=ππ

×π

Q46. From a uniform disc of radius R, a circular hole of radius R/2 is cut. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of mass of the resulting flat body.

Q58. A circular disc of radius R is removed from a bigger circular disc of radius 2R, such that the circumference of the discs coincides. The centre of mass of the new disc is α/R from the centre of the bigger disc. The value of α is

a)  1/3 b)  ½ c)  1/6 d)  1/4

Q59. A small disc of radius 2cm is cut from a disc of radius 6cm. If the distance between their centers if 3.2 cm, then what is the shift in CM of disc?

a)  0.4 cm b)  2.4 cm c)  1.8 cm d)  1.2 cm.

Q37. A circular plate of uniform thickness has a diameter of 56 cm. A circular portion of diameter 42cm is removed from the right edge of the plate with centers for both lying of the same line. The position of the CM of the remaining portion is

(a) 9cm to the right of the centre of full plate (considered as origin) (b) 9cm to the left of origin (c) 49/9cm to the left of origin (d) 49/9cm to the right of origin

Q30. Centre of mass of a disc of uniform mass and radius r, when a circular portion of radius β has been removed from it such that centre of hole is at a distance η from the centre of disc; is given by

Q31. A uniform circular disc has radius a. A s q u a r e p o r t i o n o f diagonal a is cut from it. The centre of mass of the remaining portion from the centre is

Q24. Find the CM of a uniform plate having semicircular inner and outer boundaries of radii

Q. A semicircular portion of radius ‘r’ is cut from a uniform rectangular plate as s h o w n i n f i g u r e . T h e distance of centre of mass 'C' of remaining plate, from point ‘O’ is

a)  2r/(3 – π) b)  3r/2(4 – π) c)  2r/(4 + π) d)  2r/3(4 – π)


A-3. A thin uniform wire is bent to form the two equal sides AB and AC of triangle ABC, whereAB = AC = 5 cm. The third side BC, of length 6cm, is made from uniform wire of twice the density ofthe first. The distance of centre of mass from A is :, d i rys , d l eku rkj l s f=kHkqt ABC dh nks cjkcj Hkqt k, sa AB rFkk AC cukbZ t krh gSA t gk¡ AB = AC = 5cm gS o rhl jh Hkqt k BC ft l dh yEckbZ 6cm gS] , d l eku rkj l s ft l dk ?kuRo i gys rkj dk nqxquk gS] l s cuhgSA A l s nzO; eku dsUnz dh nwjh gksxhA

(A*) 1134

cm (B) 3411

cm (C) 934

cm (D) 4511

cm

Sol.

ycm = ρ××+ρ×ρ×+ρ×

526)(2252)2(0

= 1110

Distance from A = (4 – 1110

= 1134

)

A l s nwjh = (4 – 1110

= 1134

)

A-4. A semicircular portion of radius ‘r’ is cut from a uniform rectangular plate as shown in figure. Thedistance of centre of mass 'C' of remaining plate, from point ‘O’ is :, d v) Zo‘rh; Hkkx ft l dh f=kT; k r gS dks fp=kkuql kj , d , dl eku vk; rkdkj IysV l s dkVk x; k gSA fcUnq O l scph gqbZ IysV ds nzO; eku dsUnz 'C' dh nwjh gksxhA

(A) )3(r2π− (B) )4(2

r3π− (C) )4(

r2π+ (D*) )4(3

r2π−

Sol. A1 = 2r × r = 2r2

A2 = 2r2π

x1 = 2r

x2 = π3r4

xcm =

2r–r2

3r

2r–

2rr2

22

22

ππ4

×π

×

= ]–4[3r2

2–4r

32–1r

2

3

π=

⎥⎦⎤

⎢⎣⎡ π

⎥⎦⎤

⎢⎣⎡

Q43. Find the center-of-mass coordinates xcm and ycm for the object in figure, assuming that distribution of mass is uniform.



1. Must there be mass at the centre of mass of a system? Explain.

2. Must the centre of mass of a solid body be in the interior of the body? If not, give examples.

3. A system consists of two point masses M and m(< M). The centre of mass of the system is :(a) At the middle of m and M (b) Nearer to M(c) Nearer to m (d) At the position of large mass.

4. Find the center-of-mass coordinates CMx and CMy for the object in figure, assuming that distribution ofmass is uniform.

10 m

10 m x

y

0

5. Find x and y coordinates of the centre of mass of the plate shown in figure from which a square of side 2 m iscut out. Assume that the distribution of mass is uniform.

Y

X

2m

2m6m

6m

0

6. Three identical spheres each of radius R are placed touching each other on a horizontal table as shown infigure. The x and y coordinates of the centre of mass are:

(a) (R, R) (b) (0, 0)

(c) ,2 2R R

(d) ,

3RR

. x

y

Q47. From a uniform disc of radius R, two circular sections each of radius R/4 have been removed as shown in figure. For the reference axes shown the co-ordinates of centre of mass of remaining body are




Y

X

lO

l

ll


m1 m2

m4 m3



(a)3 3,

112 112R R− −

(b)3 3,56 56

R R− −

(c)3 3,

128 128R R− −

(d)3 3,64 64

R R− −.

x

y


(- /2 0)R ,

y

x( /2 0)+R ,

RO






y

x10 m

10 m

0

Q48. Figure shows a thin uniform disc of radius R, from which a hole of radius R/2 has been cut out from left of the centre and is placed on right of the centre of disc. Find the C.M. of the resulting system.




Y

X

lO

l

ll


m1 m2

m4 m3



(a)3 3,

112 112R R− −

(b)3 3,56 56

R R− −

(c)3 3,

128 128R R− −

(d)3 3,64 64

R R− −.

x

y


(- /2 0)R ,

y

x( /2 0)+R ,

RO






y

x10 m

10 m

0

Q57. A mass M is placed on the surface of earth. Another mass M is placed at height equal to radius of earth. Find CM and CG of this system. Does the two point coincides? Why or why not?

Q31. Centre of mass and centre of gravity of a body

(a) Never coincide (b) Always coincide (c) Coincide only when gravitational field is uniform throughout (d) Have nothing to do with each other.

Q33. Two particles of mass 1kg and 3kg move towards each other under mutual force of attraction. No other force acts on them. When the relative velocity of approach of the two particles is 2m/s their CM has a velocity of 0.5 m/s and when their relative velocity of approach becomes 3m/s, the velocity of CM is 0.75m/s. Then;

(a) The above statement is correct (b) The above statement is false (c) The above statement may be correct or incorrect. (d) Data is insufficient.

Q1. A projectile is fired at a speed of 100 m/s at an angle of 370 above the horizontal. At the highest point, the projectile breaks into two parts of mass ratio 1: 3, the lighter piece coming to rest. Find the distance from the launching point to the point where the heavier piece lands. [1120m]

Q. Two men 'A' and 'B' are standing on opposite edge of a 6m long platform, which is further kept on a smooth floor. They start moving towards each other and finally meet at the midpo int o f p la t for m. F ind the displacement of platform if mass of A, B and p la t f o r m are 40kg , 60kg and 50kg respectively. [0.4m]

Q19. Mr. Verma (50kg) and Mr. Mathur (60kg) are sitting at the two extremes of a boat (40kg) 4m long standing still in water. To discuss mechanics problem, they come to the middle of boat. Neglect friction with water, how far does the boat moves on the water during the process?

Q9. Two men A (mass = M) and B (mass 2M) are standing on the two opposite ends of a boat of mass 4 M and length L on a water surface. Now A travels a distance L/4 relative to boat towards its center and B moves a distance 3L/4 relative to boat and meet A. Find the distance travelled by the boat on water till A and B meet. [5l/28]

Q22. A mass m is at rest on an inclined plane structure of mass M which is further resting on a smooth horizontal plane. Now if the mass starts moving, the position of CM of the system will

a)  Remainthesame.b)  Changealongthehorizontalc)  Changealongthever6cald)  Changealongthever6cal

whileremainsamealongthehorizontal

Q10. A block A (mass = 4M) is placed on the top of a wedge B of base length L (mass = 20M) as shown in figure. When the system is released from rest. Find the distance moved by the wedge B till the block A reaches at lowest end of wedge. All surfaces are frictionless. [L/6]

PHYSICS

"manishkumarphysics.in" 42

Problem 4. In a boat of mass 4 M and length " on a frictionless water surface. Two men A (mass = M) andB (mass 2M) are standing on the two opposite ends. Now A travels a distance "/4 relative toboat towards its center and B moves a distance 3"/4 relative to boat and meet A. Find thedistance travelled by the boat on water till A and B meet.

Solution : Let x is distance travelled by boat.Initial position of center of mass

=BABoat

BBAABoatBoat

MMMXMXMXM

��

=M2MM4

.M20.MM4 2

��

�� ""

=M7M4 "

=74"

Final position of center of mass

=^ ` ^ ` ^ `

M7

xM2xMxM4 442""" ��

=M7

Mx7M2 2M

4M �� """

=M7

xM74M11 �"

=7

x74M11 �"

since there is no horizontal force, position of center of mass remains unchanged.center of mass initially = center of mass finally

� "74

=7x74

11 �"

4" = x7411 �" � x =

285"

Problem 5. A block A (mass = 4M) is placed on the top of a wedge B of base length l (mass = 20 M) asshown in figure. When the system is released from rest. Find the distance moved by the wedgeB till the block A reaches at lowest end of wedge. Assume all surfaces are frictionless.

Solution : Initial position of center of mass

=BB

AABB

MMMXMX

��

=M24

M4.M20.XB "�= 6

X5 B "�

Final position of center of mass

=M24

Mx4M20)xX( B ��= 6

x)xX(5 B ��

since there is no horizontal force on systemcenter of mass initially = center of mass finally.5XB + " = 5XB + 5x + x" = 6x

6x

"

Q23. In the arrangement shown; the mass m starts from the top of wedge of mass M. The system was initially at rest. Assume all surfaces to be frictionless, the distance moved by M when m just reaches the floor is

Q11. A block of mass M is placed on top of a bigger block of mass 10M as shown. All the surfaces are frictionless. The system is released from rest. Find the distance moved by the bigger block at the instant the smaller block reaches the ground.

Q13. A body is falling vertically downwards under gravity breaks into two parts of unequal masses. The centre of gravity of two masses taken together shifts horizontally towards:

a)  Heavier piece b)  Lighter piece c)  Does not shift horizontally d)  Depends upon the vertical velocity at

the time of breaking.

Q21. The balloon, the light rope and the monkey shown in the figure are all at rest in air. If the monkey reaches the top of the rope, by what distance does the balloon descend?

Q25. A shell at rest at origin explodes into three fragments of masses 1 kg, 2 kg and m kg. The fragments of masses 1 kg and 2 kg fly off with speeds 12 m/s along x-axis and 8 m/s along y- axis respectively. If m kg flies off with speed 40 m/s then find the total mass of the shell.

[3.5 kg]

Q26. Two bodies of masses 1 kg and 2 kg are moving in two perpendicular direction with velocities 1 m/s and 2 m/s as shown in figure. The velocity of the centre of mass (in magnitude) of the system will be;

a)  3 m/s b)  1.67 m/s c)  1.5 m/s d)  1.37 m/s



1. Two bodies of masses 1 kg and 2 kg are moving in two perpendicular direction with velocities 1 m/s and 2m/s as shown in figure. The velocity of the centre of mass (in magnitude) of the system will be;

(a) 3 m/s

(b) 1.67 m/s

(c) 1.5 m/s

(d) 1/37 m/s.

1 kg

2 kg

2 m/s

1 m/s

2. Two particles A and B initially at rest, move towards each other mutual force of attraction. At the instantwhen the speed of A is V and the speed of B is 2V, the speed of the centre of mass of the system is:(a) 3 V (b) V(c) 1.5 V (d) zero

3. A particle of mass 4 m which is at rest explodes into three fragments. Two of fragments, each of mass m arefound to move with a speed v each in mutually perpendicular directions. The total energy released in theprocess is;

(a)12 mv² (b) mv²

(c)32 mv² (d)

52 mv².

4. Two particles having position vectors 1 (3 5 )r i j= +$ $$ meter and 2 ( 5 3 )r i j= − −

$ $$ meter are moving withvelocities 1 (4 3 )v i j= +

$ $$ m/s and 2 ( 7 )v ai j= +$ $$ m/s. If they collide after 2 second, the value of a;

(a) 2 (b) 4(c) 6 (d) 8.

5. A bomb at rest explodes into large number of tiny fragments. Then:

(a) the momentum of all the fragments is zero(b) the momentum of all the fragments increases(c) the K. E. of all the fragment remain zero(d) the K. E. of all the fragment is more than zero.

6. A shell is fired from a cannon with a velocity v(m/s) at an angle θ with the horizontal direction. At the highestpoint in its path it explodes into two pieces of equal mass. One of the pieces retraces its path to the cannonand the speed (m/s ) of the other piece immediately after the explosion is :

(a) 3 cosv θ (b) 2 cosv θ

(c)3 cos2

v θ (d)3 cos2

v θ

7. A body A of mass M while falling vertically downward under gravity breaks into two parts; a body B of

mass 3M

and a body C of mass 2

3M

. The centre of mass of bodies B and C taken together shiftscompared to that of body a towards:(a) Body (b) Body B(c) Depends on height of breaking (d) Does not shift.

Q27. Two particles A and B initially at rest, move towards each other mutual force of attraction. At the instant when the speed of A is V and the speed of B is 2V, the speed of the centre of mass of the system is:

(a) 3V (b) V (c) 1.5 V (d) Zero

Q28. A body A of mass M while falling vertically downward under gravity breaks into two parts; a body B of mass M/3 and a body C of mass 2M/3. The centre of mass of bodies B and C taken together shifts compared to that of body a towards:

(a) Body (b) Depends on height of breaking (c) Body B (d) Does not shift.

Q30. A boy is standing at the stern (back) of a boat that is 8.0 m long. There is no friction between the boat and the water. The boy has a mass of 63 kg and the boat has a mass of 780 kg. The bow (front) of the boat is touching a dock and the force-and-raft axis of the boat is perpendicular to the dock. The boy walks from the stern of the boat to the bow. When he reaches the bow, his distance from the dock is (a) 7.6 m (b) 0.65 m (c) 0.51 m (d) 0.59 m (e) 1.3 m.

Q. A uniform sphere is placed on a smooth horizontal surface and a horizontal force F is applied on it at a distance h above the centre. The acceleration of the centre of mass of the sphere

a)  Is maximum when h = 0 b)  Is maximum when h = R

Is maximum when h = R/2 c)  Is independent of h

Q. A ball moves horizontally in a closed box making several collisions with the walls. The box is kept on a smooth horizontal surface. The velocity of the centre of mass during the motion of the ball:

a)  Of the box remains constant Of the box plus the ball system remains constant Depends on value of coefficient of restitution. Of the ball relative to the box remains constant

Q. A thin uniform sheet of metal of uniform thickness is cut into the shape bounded by the line x = a and y = ± kx2, as shown. Find the coordinates of the centre of mass.


Sol.

Let areal density is ρekuk i ‘"Bh; ?kuRo ρ gS

MC = ρ×⎟⎠⎞

⎜⎝⎛π

2

2a

Ms = a2 × ρ xS = a XC = 0

2ρ+π

ρ

ρ=

a4a

)a(aX 2

2

cm

)4(a4π+

right of the disc centre pdrh ds dsUnz l s nka; h vksj

A 8. A thin uniform sheet of metal of uniform thickness is cut into the shape bounded by the line x = a andy = ± k

x2, as shown. Find the coordinates of the centre of mass.

/kkrq dh , d i ryh , d l eku eksVkbZ okyh i jr dks fp=kkuql kj js[kk x = a rFkk y = ± k x2, l s i fjc) vkd‘f~Ùk ds vUrxZr

dkVk t krk gSA nzO; eku dsUnz ds funsZ’kkad crkb; sA

Ans. ⎟⎠⎞

⎜⎝⎛ O,a

43

Sol.

length of the shaded region Nk; kafdr Hkkx dh yackbZ = 2y = 2kx2

dm = 2y dx × ρdm = 2kx2 × ρ dx

M = ∫ ρ=∫a

o

2a

odxkx2dm =

3ak2 3ρ

Xcm =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛ρ

=

∫

∫ ρ=

∫

∫

3ak2

4ak2

dm

dxkx2

dm

dmx

3

4

a

0

a

0

3

a

0

a

0 Xcm = 4a3

By symmetry the y-coordinate of the shown plate is zero.l efer rk l s nh xbZ IysV dk y funsZ’kkad ’kwU; gSA

Q21. Two particles of equal mass have initial velocities 2i and 2j in m/s. . First particle has constant acceleration of (i + j) m/s2while the acceleration of the second particle is always zero. The centre of mass of the two particles moves in

a) Circle b) Parabola c) Ellipse d) Straight line

Q. A body of mass 1 kg moving in the x-direction, suddenly explodes into two fragments of mass 1/8kg and 7/8 kg. An instant later, the smaller fragment is 0.14 m above the x-axis. The position of the heavier fragment is

a)  1/50 m above x-axis. b)  1/50m below x-axis. * c)  7/50m above x-axis d)  7/50m below x-axis

Q. A block of mass M with a semicircular track of radius R rests on a horizontal frictionless surface. A uniform cylinder of radius r and mass m is released from rest from the top point A. The cylinder slips on the semicircular frictionless track. The distance travelled by the block when the cylinder reaches the point B is:


Sol. An external force of 3mω2R is required which can act anywhere on system.3mω2R , d ckg; cy vko’ ; d gS t ks fudk; i j dgh Hkh dk; Zj r gks l drk gSA

5. A block of mass M with a semicircular track of radius R rests on a horizontal frictionless surface. Auniform cylinder of radius r and mass m is released from rest from the top point A. The cylinder slipson the semicircular frictionless track. The distance travelled by the block when the cylinder reachesthe point B is :M nzO; eku dk , d fi .M ft l esa R f=kT; k dk v}Zo‘Ùkkdkj i Fk gS] , d {kSfr t ?k"kZ.kj fgr ry i j fojkekoLFkk esa j [ kkgSA , d l eku csyu ¼nzO; eku m, f=kT; k r½ dks Åi jh fcUnq A l s fojkekoLFkk l s NksM+k t krk gSA csyu ?k"kZ.kj fgrv}Zo‘Ùkkdkj i Fk i j fQl yrk gSA t c csyu fcUnq B rd i gq¡prk gS rks xqVds }kjk r ; dh xbZ nwjh gS &

(A) mM

)rR(M+−

(B*) mM

)rR(m+−

(C) M

R)mM( +(D) none dksbZ ugha

Sol. when cylinder reaches pt B. t c xsan fcUnq B i j i gqaprh gS rksthen block get shifted by x CykWd x nwjh l s foLFkkfi r gks t krk gS∴ but since than there is no ext fdUrq dksbZ cká cy ugh gSaforce therefore com remain at its position vr% nzO; eku dsUnz fLFkj jgsxk[(R–r) – x]m = Mx

∴ x = mM

)r–R(m+

6. In the above question, the velocity of the block when the cylinder reaches point (B) is :mi jksDr i z’u esa t c csyu fcUnq B i j i gq¡prk gS rc fi .M dk osx gS %

(A) )mM(M)rR(g2

M+−

(B) )mM(m)rR(g2

m+−

(C*) )mM(M)rR(g2

m+−

(D) )mM(M)rR(g2

M++

Sol. Using momentum conservation l aosx l aj{k.k l s

MV = mv V = M

mv......(i)

using energy conservation equation Åt kZ l aj{k.k l s

mg(R–r) = 21

mv2 + 21

Mv2 ......(ii)

on solving we get gy djus i j i zkIr gksxk v = )m–m(M)r–R(g2

m .

7. A uniform thin rod of mass M and Length L is standing vertically along the y-axis on a smoothhorizontal surface, with its lower end at the origin (0,0). A slight disturbance at t = 0 causes the lowerend to slip on the smooth surface along the positive x-axis, and the rod starts falling. The accelerationvector of centre of mass of the rod during its fall is :

[ R is reaction from surface], d l e: i i r yh M nzO; eku , oa L yEckbZ dh NM+ y-v{k ds vuqfn’k fpdus {kSfr t r y i j Å/okZ/kj [ kM+hgSA bl dk fupyk fl j k ewy fcUnq (0,0) i j gSA t = 0 i j y?kq fo{kksHk l s gh bl dk fupyk fl j k fpduh l r gi j /kukRed x-v{k dh vksj fQl yr k gs r Fkk NM+ fxj uk i zkj EHk dj r h gS fxj r s l e; bl ds nzO; eku dsUnzdk Roj .k l fn’k gksxkA [ R l r g l s i zfr fØ; k cy gSA]

(A*) M

RgMaCM

rrr +

= (B) M

RgMaCM

rrr −

= (C) RgMaCM

rrr−= (D) None of these buesa l s dksbZ

ughaA

x =m R− r( )M +m

Q34. In a vertical plane inside a smooth hollow thin tube a block of same mass as that of tube is released as shown in figure. When it is slightly disturbed it moves towards right. By the time the block reaches the right end of the tube, displacement of the tube will be (where ‘R’ is mean radius of tube). Assume that the tube remains in vertical plane.


8. In a vertical plane inside a smooth hollow thin tube a block of same mass as that of tube is released asshown in figure. When it is slightly disturbed it moves towards right. By the time the block reaches theright end of the tube, displacement of the tube will be (where ‘R’ is mean radius of tube). Assume thatthe tube remains in vertical plane.Å/okZ/kj ry esa [ kks[ kyh ufydk esa l eku nzO; eku dk CykWd fp=kkuql kj NksM+k t krk gS t c ; g ufydk ds , d fl jsi j i gq¡prk gS rc ufydk dk foLFkki u gksxk (‘R’ ufydk dh vkSl r f=kT; k gS) ekuks ufydk Å/okZ/kj ry esa cuhjgrh gSA

(A) πR2

(B) πR4

(C*) 2R

(D) R

Sol. Since there is no ext. force on systempwafd fudka; i j dksbZ cká cy ugh gS vr%m (R – x) + m (–x) = 0x = R/2.

Alternate : Let the tube displaced by x towards left, then ; ekuk ufydk x foLFkkfi r gksrh gSA rc &

mx = m (R – x) ⇒ x = 2R

9. Two men ‘A’ and ‘B’ are standing on a plank. ‘B’ is at the middle of the plank and ‘A’ is at the left endof the plank. Bottom surface of the plank is smooth. System is initially at rest and masses are asshown in figure. ‘A’ and ‘B’ start moving such that the position of ‘B’ remains fixed with respect toground and ‘A’ meets ‘B’. Then the point where A meets B is located at :nks vkneh ‘A’ rFkk ‘B’ IykUd i j [ kM+s gSA ‘B’ IykUd ¼i V~Vs½ ds e/; esa gS rFkk ‘A’ IykUd ds cka; s fl js i j gSA IykUd¼i V~Vk½ dh l rg fpduh gSA fudk; i zkjEHk esa fLFkj gS rFkk nzO; eku fp=kkuql kj gSA ‘A’ rFkk ‘B’ bl i zdkj xfr dj rsgS fd ‘B’ (t ehu ds l ki s{k) fLFkj jgrk gS rFkk ‘A’, ‘B’ l s feyrk gSA rc og fcUnq t gk¡ A, B l s feyrk gS dh fLFkfrgksxh &

A B40 kg 60 kg

40 kg

120 cm

smooth( )fpduh

(A) the middle of the plank (B) 30 cm from the left end of the plank(C*) the right end of the plank (D) None of these(A) IykUd ¼i V~Vk½ ds e/; esa (B) IykUd ¼i V~Vk½ ds cka; s fl js l s 30 cm dh nwjh i j(C*) IykUd ¼i V~Vk½ ds nkfgus fl js i j (D) buesa l s dksbZ ugha

[R/2]

Q. Two blocks of masses 2 kg and 1 kg respectively are tied to the ends of a string which passes over a light frictionless pulley. The masses are held at rest at the same horizontal level and then released. The distance traversed by centre of mass in 2 s is (g = 10 m/s2)

[2.22m]

Q. Mass is non-uniformly distributed on the circumference of a ring of radius a and centre at origin. Let b be the distance of centre of mass of the ring from origin. Then,

(a) b = a (b) 0 ≤ b ≤ a * (c) b < a (d) b > a

Q. A block A slides over an another block B which is placed over a smooth inclined plane as shown in figure. The coefficient of friction between the two blocks A and B is µ. Mass of block B is two times the mass of block A. The acceleration of the centre of mass of two blocks is________

Q. Two blocks of equal mass are tied with a light string, which passes over a massless pulley as shown in figure. The magnitude of acceleration of centre of mass of both the blocks is (neglect friction everywhere)

Q. A system of two blocks A & B and a wedge C are released from rest as shown. Masses of the blocks and the wedge are m, 2 m and 2 m respectively. The displacement of wedge C when block B slides down the plane a distance 10 cm is (neglect friction)

Q. A small sphere of radius R held against the inner surface of a smooth spherical shell of radius 6R as shown in figure. The masses of the shell and small spheres are 4M and M respectively. This arrangement is placed on a smooth horizontal table. The small sphere is now released. The x-coordinate of the centre of the shell when the smaller sphere reaches the other extreme position is

Q. Two blocks A and B each of equal masses m are released from the top of a smooth fixed wedge as shown in the figure. Find the magnitude of the acceleration of the centre of mass of the two blocks.

Q. Figure shows a fixed wedge on which two blocks of masses 2 kg and 3 kg are placed on its smooth inclined surfaces. When the two blocks are released from rest, find the acceleration of centre of mass of the two blocks.

Q. A machinist starts with three identical square plates but cuts one corner from one of them, two corners from the second and three corners from the third. Rank the three according to the x-coordinate of their centre of mass, from smallest to

Q. In the figure shown a hole of radius 2 cm is made in semicircular disc of radius 6π at a distance 8 cm from the centre C of the disc. The distance of the centre of mass of this system from point C is _______

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1. A semicircular portion of radius 'r' is cut from auniform rectangular plate as shown in figure. Thedistance of centre of mass 'C' of remaining plate, frompoint 'O' is

2r

r

C O

(A) 2

3r

( ñ ) (B) 3

2 4r

( ñ )

(C) 24r

( ) (D) 2

3 4r

( ñ )

2. From a circle of radius a, an isosceles right angledtriangle with the hypotenuse as the diameter of thecircle is removed. The distance of the centre of gravityof the remaining position from the centre of the circle is

(A) 3( ñ 1)a (B) ( ñ ) 1

6a

(C) a

3 1( ñ ) (D) a

3 1( )

3. In the figure shown a hole of radius 2 cm is made insemicircular disc of radius 6 at a distance 8 cm fromthe centre C of the disc. The distance of the centre ofmass of this system from point C is

2cm8cm

(A) 4 cm (B) 8 cm(C) 6 cm (D) 12 cm

4. Centre of mass of two thin uniform rods of samelength but made up of different materials & kept asshown, can be, if the meeting point is the origin ofco-ordinates

L

Lx

y

(A) (L/2, L/2) (B) (2L/3, L/2)

(C) (L/3, L/3) (D) (L/3, L/6)

5. A man of mass M stands at one end of a plank oflength L which lies at rest on a frictionless surface.The man walks to other end of the plank. If the mass

of the plank is M3 , then the distance that the man

moves relative to ground is :

(A) 34L

(B) L4

(C) 45L

(D) L3

6. A particle of mass 3m is projected from the groundat some angle with horizontal. The horizontal range isR. At the highest point of its path it breaks into twopieces m and 2m. The smaller mass comes to rest andlarger mass finally falls at a distance x from the pointof projection where x is equal to

(A) 34R

(B) 32R

(C) 54R

(D) 3R

7. A man weighing 80 kg is standing at the centre ofa flat boat and he is 20 m from the shore. He walks 8m on the boat towards the shore and then halts. Theboat weight 200 kg. How far is he from the shore atthe end of this time?(A) 11.2 m (B) 13.8 m(C) 14.3 m (D) 15.4 m

8. Two particles having mass ratio n : 1 areinterconnected by a light inextensible string thatpasses over a smooth pulley. If the system is released,then the acceleration of the centre of mass of thesystem is :

(A) (n ñ1)2 g (B) nn

g

11

2

ñ

(C) nn

gñ 11

2

(D)

nn

g

11ñ

Question No. 9 to 10 (2 questions)A uniform chain of length 2L is hanging in equilibriumposition, if end B is given a slightly downwarddisplacement the imbalance causes an acceleration.Here pulley is small and smooth & string is inextensible

A B

Exercise - I Objective Problems

a)  4 cm b)  8 cm* c)  6 cm d)  12 cm

IITJEE – MAINS & ADVANCED AIIMS / AIPMT Concept of Centre of Mass - Physics...

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