Physics 4A Winter 2016 Final Exam

Name:

Mar 22, 2016

Please show your work! Answers are not complete without clear reasoning. When askedfor an expression, you must give your answer in terms of the variables given in the questionand/or fundamental constants.

Answer as many questions as you can. Do not forget to include appropriate units whengiving a number as an answer. Calculators are allowed. If you detach any pages from thetest, please write your name on every page.

ConstantsG = 6.67 × 10−11 N m2 kg−2

g = 9.8 m s−2 (If you like you can use g = 10 m s−2, but make your choice clear.)

Equationsxf = xi + vit+ 1

2at2

vf = vi + at

vavg =vi+vf

2

v2f = v2i + 2a∆x

xf = xi + vavgt

ω = 2πT

v = rω

at = rα

ac = v2

r

vf = vi + ve ln(mi

mf

)Thrust = ve

dmdt

rCM = 1Mtot

∑imiri

I =∑

imir2i

I ′ = ICM +MD2

T = 2π√

mk

T = 2π√

Lg

K = 12mv2

Ug = mgy

Us = 12kx2

UG = −Gm1m2

r

W =∫τ · dθ =

∫F · ds

P = τ · ω = F · vFx = − dU

dx

R = −bvv(t) = vT (1 − e−bt/m)

R = 12DρAv2

v(t) = vT tanh(

gvTt)

dAdt

= L2Mp

T 2 =(

4π2

GM

)a3

p = mv

F = dpdt

τ = r× F

L = r× p

τ = dLdt

I =∫F(t) dt

∆L =∫τ dt

L = Iω

L = mvR

ωp = Mg rCM

Iω

Fk = µkn

Fs,max = µsn

F = −kxFG = −Gm1m2

r2r

E = −Gm1m2

2a

1

1. Two students are on a balcony a distance h above the street. One student throws aball vertically downward at a speed vi; at the same time, the other student throws aball vertically upward at the same speed. Answer the following symbolically in termsof vi, g, h, and t.

(a) What is the time interval between when the first ball strikes the ground and thesecond ball strikes the ground? [4pts]

(b) Find the velocity of each ball as it strikes the ground. [3pts]

(c) How far apart are the balls at a time t after they are thrown and before theystrike the ground? [3pts]

(d) Sketch a height-versus-time graph for both balls on the same axes. [2pts]

(e) Sketch a velocity-versus-time graph for both balls on the same axes. [2pts]

2

2. A small particle of mass m is pulled to the top of a frictionless half-cylinder (of radiusR) by a light cord that passes over the top of the cylinder as shown. The force F issuch that the particle moves at a constant speed v.

(a) What is the normal force on the particle when it is in the position shown? [3pts]

(b) Show that F = mg cos θ. [2pts]

(c) By directly integrating W =∫F · dr, find the work done in moving the particle

at constant speed from the bottom to the top of the half-cylinder. [4pts]

(d) What is the net work done on the particle? [2pts]

206 Chapter 7 Energy of a System

27. When different loads hang on a spring, the spring stretches to different lengths as shown in the follow-ing table. (a) Make a graph of the applied force versus the extension of the spring. (b) By least-squares fit-ting, determine the straight line that best fits the data. (c) To complete part (b), do you want to use all the data points, or should you ignore some of them? Explain. (d) From the slope of the best-fit line, find the spring constant k. (e) If the spring is extended to 105 mm, what force does it exert on the suspended object?

F (N) 2.0 4.0 6.0 8.0 10 12 14 16 18 20 22

L (mm) 15 32 49 64 79 98 112 126 149 175 190

28. A 100-g bullet is fired from a rifle having a barrel 0.600 m long. Choose the origin to be at the location where the bullet begins to move. Then the force (in newtons) exerted by the expanding gas on the bullet is 15 000 1 10 000x  2 25 000x2, where x is in meters. (a) Determine the work done by the gas on the bullet as the bullet travels the length of the barrel. (b) What If? If the barrel is 1.00 m long, how much work is done, and (c) how does this value compare with the work cal-culated in part (a)?

29. A force FS

5 14x i 1 3y j 2 , where FS

is in newtons and x and y are in meters, acts on an object as the object moves in the x direction from the origin to x 5 5.00 m. Find the work W 5 e F

S? d rS done by

the force on the object.

30. Review. The graph in Figure P7.30 specifies a functional relationship between the two vari-ables u and v. (a) Find eb

a u dv. (b) Find ea

b u dv.

(c) Find eb

a v du.

Section 7.5 Kinetic Energy and the Work–Kinetic Energy Theorem 31. A 3.00-kg object has a velocity 16.00 i 2 2.00 j 2 m/s.

(a) What is its kinetic energy at this moment? (b) What is the net work done on the object if its velocity changes to 18.00 i 1 4.00 j 2 m/s? (Note: From the definition of the dot product, v2 5 vS ? vS .)

32. A worker pushing a 35.0-kg wooden crate at a constant speed for 12.0 m along a wood floor does 350 J of work by applying a constant horizontal force of magnitude F on the crate. (a) Determine the value of F. (b) If the worker now applies a force greater than F, describe the subsequent motion of the crate. (c) Describe what would happen to the crate if the applied force is less than F.

33. A 0.600-kg particle has a speed of 2.00 m/s at point ! and kinetic energy of 7.50 J at point ". What is (a) its kinetic energy at !, (b) its speed at ", and (c) the net work done on the particle by external forces as it moves from ! to "?

Q/C

–4

0

u (N)

v (cm)10 20 30

a

b

4

8

Figure P7.30

W

W

Q/CAMT

W

(b) How much work does the archer do on the string in drawing the bow?

20. A light spring with spring constant 1 200 N/m is hung from an elevated support. From its lower end hangs a second light spring, which has spring constant 1 800 N/m. An object of mass 1.50 kg is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as in series.

21. A light spring with spring constant k1 is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant k2. An object of mass m is hung at rest from the lower end of the sec-ond spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system.

22. Express the units of the force constant of a spring in SI fundamental units.

23. A cafeteria tray dispenser supports a stack of trays on a shelf that hangs from four identical spiral springs under tension, one near each corner of the shelf. Each tray is rectangular, 45.3 cm by 35.6 cm, 0.450 cm thick, and with mass 580 g. (a) Demonstrate that the top tray in the stack can always be at the same height above the floor, however many trays are in the dispenser. (b) Find the spring constant each spring should have for the dispenser to function in this convenient way. (c) Is any piece of data unnecessary for this determination?

24. A light spring with force constant 3.85 N/m is com-pressed by 8.00 cm as it is held between a 0.250-kg block on the left and a 0.500-kg block on the right, both rest-ing on a horizontal surface. The spring exerts a force on each block, tending to push the blocks apart. The blocks are simultaneously released from rest. Find the acceleration with which each block starts to move, given that the coefficient of kinetic friction between each block and the surface is (a) 0, (b) 0.100, and (c) 0.462.

25. A small particle of mass m is pulled to the top of a frictionless half-cylinder (of radius R) by a light cord that passes over the top of the cyl-inder as illustrated in Figure P7.25. (a) Assum-ing the particle moves at a constant speed, show that F 5 mg cos u. Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all times. (b) By directly integrating W 5 e F

S? d rS, find

the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder.

26. The force acting on a particle is Fx 5 (8x 2 16), where F is in newtons and x is in meters. (a) Make a plot of this force versus x from x 5 0 to x 5 3.00 m. (b) From your graph, find the net work done by this force on the particle as it moves from x 5 0 to x 5 3.00 m.

S

S

R

u

FS

m

Figure P7.25

S

3

3. A pendulum bob of mass m is attached to a light string of length L. The string makesa small angle θi with the vertical and then the bob is released with an initial velocityv as in the diagram. (Assume there is no friction or air resistance.)

g

θ

m

L

v

(a) What is the period of the bob’s oscillation? [1pt]

(b) What is the bob’s initial angular speed? [1pt]

(c) Assume that the bob’s motion can be described by θ(t) = θmax cos(ω0t+φ). Findexpressions for ω0, θmax, and φ. [8pts]

4

4. Consider an object that rolls down an incline.

θ

(a) Determine the acceleration of the center of mass of a uniform solid disk rollingdown an incline making angle θ with the horizontal. [5pts]

(b) What is the minimum coefficient of friction required to maintain pure rollingmotion for the disk? [4pts]

(c) Consider a hollow cylinder of mass m, inner radius r1, and outer radius r2 anduniform density. Starting from the expression for the moment of inertia of auniform disk, show that the moment of inertia of the hollow cylinder is [5pts]

I =1

2m(r21 + r22)

(d) Find the acceleration of this hollow cylinder down the same slope assuming itrolls without slipping. Compare the acceleration found in part (a) with that ofthe hollow cylinder. Is one always larger than the other, or does it depend on themass and radius of the disk and cylinder which will be larger? [6pts]

5

5. A uniform sign of mass M and width 2L hangs from a uniform horizontal beam ofmass m hinged at the wall and supported by a cable as shown. Determine

(a) the tension in the cable [4pts]

(b) and the components of the reaction force exerted by the wall on the beam [5pts]

in terms of M , m, g, d, L, and θ.

384 Chapter 12 Static Equilibrium and Elasticity

form, weighs 200 N, and is 6.00 m long, and it is sup-ported by a wire at an angle of u 5 60.0°. The basket weighs 80.0 N. (a) Draw a force diagram for the beam. (b) When the bear is at x 5 1.00 m, find the tension in the wire supporting the beam and the components of the force exerted by the wall on the left end of the beam. (c) What If? If the wire can withstand a maxi-mum tension of 900 N, what is the maximum distance the bear can walk before the wire breaks?

44. The following equations are obtained from a force diagram of a rectangular farm gate, supported by two hinges on the left-hand side. A bucket of grain is hang-ing from the latch.

2A 1 C 5 0

1B 2 392 N 2 50.0 N 5 0

A(0) 1 B(0) 1 C(1.80 m) 2 392 N(1.50 m)

2 50.0 N(3.00 m) 5 0

(a) Draw the force diagram and complete the statement of the problem, specifying the unknowns. (b) Deter-mine the values of the unknowns and state the physical meaning of each.

45. A uniform sign of weight Fg and width 2L hangs from a light, horizontal beam hinged at the wall and sup-ported by a cable (Fig. P12.45). Determine (a) the tension in the cable and (b) the components of the reaction force exerted by the wall on the beam in terms of Fg , d, L, and u.

u

d

2L

Lulu and Lisa’s Cafe

Figure P12.45 46. A 1 200-N uniform boom at f 5 658 to the vertical is

supported by a cable at an angle u 5 25.08 to the hori-zontal as shown in Figure P12.46. The boom is pivoted at the bottom, and an object of weight m 5 2 000 N hangs from its top. Find (a) the tension in the support cable and (b) the components of the reaction force exerted by the floor on the boom.

!34 !

u

m

f

Figure P12.46

47. A crane of mass m1 5 3 000 kg supports a load of mass m2 5 10 000 kg as shown in Figure P12.47. The crane

S

is the new tension T2 in the rod? (c)  The rod will then be how much longer than its unstressed length? (d)  When the concrete was poured, the rod should have been stretched by what extension distance from its unstressed length? (e)  Find the required original tension T1 in the rod.

41. The arm in Figure P12.41 weighs 41.5 N. The gravita-tional force on the arm acts through point A. Deter-mine the magnitudes of the tension force F

St in the

deltoid muscle and the force FS

s exerted by the shoul-der on the humerus (upper-arm bone) to hold the arm in the position shown.

FsS

FtS

FgS

29.0 cm8.00 cm

12.0!

OA

Figure P12.41

42. When a person stands on tiptoe on one foot (a strenu-ous position), the position of the foot is as shown in Figure P12.42a. The total gravitational force F

Sg on the

body is supported by the normal force nS exerted by the floor on the toes of one foot. A mechanical model of the situation is shown in Figure P12.42b, where T

S is

the force exerted on the foot by the Achilles tendon and R

S is the force exerted on the foot by the tibia.

Find the values of T, R, and u when Fg 5 700 N.

18.0 cm25.0 cm

15.0!

Tibia

Achillestendon

a b

nS

RS T

Su

Figure P12.42

43. A hungry bear weighing 700  N walks out on a beam in an attempt to retrieve a basket of goodies hanging at the end of the beam (Fig. P12.43). The beam is uni-

BIO

BIO

AMT

u

x

Goodies

Figure P12.43

6

6. A comet of mass m moves about a star in an elliptical orbit, with its closest approachto the star being r1 and its greatest distance r2. The comet’s speed at closest approachis v1.

(a) Consider the angular momentum of the comet about the star. What is the changein angular momentum as the comet moves from its closest point to the star to itsfurthest point from the star? (Argue by considering the net torque on the comet.)[3pts]

(b) What is its speed when it is farthest from the star? [3pts]

(c) What is the work done on the comet by the force of gravitational attraction asthe comet moves from its closest point to the star to its furthest point from thestar? [3pts]

(d) Just for this part of the question, assume the mass of the star is M . What is theperiod of the comet’s orbit? [2pts]

7

7. A projectile of mass m is launched with an initial velocity vi making an angle θ withthe horizontal as shown. The projectile moves in the gravitational field of the Earth.

356 Chapter 11 Angular Momentum

18. A counterweight of mass m 5 4.00 kg is attached to a light cord that is wound around a pulley as in Fig-ure P11.18. The pulley is a thin hoop of radius R  5 8.00 cm and mass M 5 2.00 kg. The spokes have neg-ligible mass. (a)  What is the magnitude of the net torque on the system about the axle of the pulley? (b) When the counterweight has a speed v, the pulley has an angular speed v 5 v/R. Determine the mag-nitude of the total angular momentum of the system about the axle of the pulley. (c) Using your result from part (b) and tS 5 d L

S/dt, calculate the acceleration of

the counterweight.

m

R

M

Figure P11.18

19. The position vector of a particle of mass 2.00 kg as a function of time is given by rS 5 16.00 i 1 5.00t j 2 , where rS is in meters and t is in seconds. Determine the angular momentum of the particle about the origin as a function of time.

20. A 5.00-kg particle starts from the origin at time zero. Its velocity as a function of time is given by

vS 5 6t2 i 1 2t j

where vS is in meters per second and t is in seconds. (a) Find its position as a function of time. (b) Describe its motion qualitatively. Find (c) its acceleration as a function of time, (d) the net force exerted on the par-ticle as a function of time, (e) the net torque about the origin exerted on the particle as a function of time, (f) the angular momentum of the particle as a func-tion of time, (g) the kinetic energy of the particle as a function of time, and (h) the power injected into the system of the particle as a function of time.

21. A ball having mass m is fas-tened at the end of a flagpole that is connected to the side of a tall building at point P as shown in Figure P11.21. The length of the flagpole is ,, and it makes an angle u with the x axis. The ball becomes loose and starts to fall with accelera-tion 2g j. (a)  Determine the angular momentum of the ball about point P as a function of time. (b) For what physical reason does the angular momentum change? (c) What is the rate of change of the angular momen-tum of the ball about point P ?

WAMT

M

Q/C

m

!

P

u

Figure P11.21

Q/CS

of the particle about the origin when its position vector is rS 5 x i 1 y j.

14. Heading straight toward the summit of Pike’s Peak, an airplane of mass 12 000 kg flies over the plains of Kan-sas at nearly constant altitude 4.30 km with constant velocity 175 m/s west. (a) What is the airplane’s vector angular momentum relative to a wheat farmer on the ground directly below the airplane? (b) Does this value change as the airplane continues its motion along a straight line? (c) What If? What is its angular momen-tum relative to the summit of Pike’s Peak?

15. Review. A projectile of mass m is launched with an ini-tial velocity vSi making an angle u with the horizontal as shown in Figure P11.15. The projectile moves in the gravitational field of the Earth. Find the angular momentum of the projectile about the origin (a) when the projectile is at the origin, (b) when it is at the high-est point of its trajectory, and (c) just before it hits the ground. (d) What torque causes its angular momen-tum to change?

O R

vxi i

u

my

x

viS

v2S

v1S

!

Figure P11.15

16. Review. A conical pendulum consists of a bob of mass m in motion in a cir-cular path in a horizontal plane as shown in Figure P11.16. During the motion, the supporting wire of length , maintains a constant angle u with the vertical. Show that the magnitude of the angular momentum of the bob about the vertical dashed line is

L 5 am2g ,3 sin4 ucos u

b1/2

17. A particle of mass m moves in a circle of radius R at a constant speed v as shown in Figure P11.17. The motion begins at point Q at time t 5 0. Determine the angular momentum of the particle about the axis perpendicu-lar to the page through point P as a function of time.

Q/CS

!

m

u

Figure P11.16

S

S

mR

y

xQP

vS

Figure P11.17 Problems 17 and 32.

(a) What are the components of the velocity of the projectile as a function of time? [2pts]

(b) Find the time-of-flight of the projectile. (Show a derivation.) [3pts]

(c) Find the range of the projectile, R. (Show a derivation.) [3pts]

(d) Find the angular momentum of the projectile about the origin as a function oftime, L(t), between its launch and landing. [5pts]

(e) Check your expression in the previous part is correct for the instant just before ithits the ground. Do this by plugging your time-of-flight into your expression forL(t) and confirming that it is equal to the angular momentum about the originjust as it will strike the ground using your range result. [4pts]

(f) What torque causes its angular momentum to change? [1pt]

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11

Moments of Inertia

All objects listed here have mass M .

Thin rod, length L, axis through CM perpendicular to rod: I = 112ML2

Thin rod, length L, axis through endpoint of rod, perpendicular to rod: I = 13ML2

Solid sphere, radius R, axis through CM: I = 25MR2

Cylinder or disc, radius R, axis through CM: I = 12MR2

Thin ring, radius R, axis through CM: I = MR2

Trigonometric Identities

sin2 θ + cos2 θ = 1

sin(2θ) = 2 sin(θ) cos(θ).

cos(2θ) = cos2 θ − sin2 θ

sin(α± β) = sinα cos β ± cosα sin β

cos(α± β) = cosα cos β ∓ sinα sin β

cosα cos β = 12[cos(α− β) + cos(α + β)]

sinα sin β = 12[cos(α− β) − cos(α + β)]

sinα cos β = 12[sin(α + β) + sin(α− β)]

sin(θ + π

2

)= cos θ

cos(θ + π

2

)= − sin θ

sec θ := 1cos θ

csc θ := 1sin θ

cot θ := 1tan θ

12