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PHY151H1: MIDTERM 2010 FRIDAY, 29 OCT 2010, 6:10 PM-8PM

Instructions: Complete all 6 questions. All questions have equal weight. There is an aid sheet with possibly useful formulas attached. Make sure to show all your work, label all diagrams and include units and directions where necessary.

1. A wheel’s angular velocity from t = 0.0 s to t = 2.0 s is shown in Figure 1. a. What is the wheel’s angular acceleration at t = 0.0 s? b. How many revolutions does the wheel complete from t = 0.0 s to t = 2.0 s? c. If the centripetal acceleration of a point on the edge of the wheel at t = 1.5 s is 45 m/s2,

what is the wheel’s radius?

2. A copy of Decline and Fall by Evelyn Waugh lies exactly in the middle of an otherwise empty 6.1m long bookshelf in Robarts Library. The coefficient of static friction for the book on the shelf is µS = 11/60 and the shelf has been polished (so kinetic friction may be neglected). A tall, disgruntled librarian slowly lifts up the one end of the bookshelf at a constant rate, so that it pivots upward, anchored at the opposite end (see Figure 2). After 5.5 seconds, she sees the book begin to slide down the tilted shelf. She continues to raise her end of the shelf at the same rate, gleefully watching until the book falls onto the floor.

a. What is the speed at which the librarian raised the shelf? b. Draw a graph of the book’s acceleration as a function of time. c. How long does it take the book to fall off the shelf (measured from the time it starts

moving)?

3. The acceleration of the newly discovered peon particle in the University of Toronto’s linear accelerator (in which motion is restricted to a straight line) is given by

!

a(t) = 2t + b where b is some constant. Its velocity at t = 0 seconds is v=1 m/s. If x(0)=0, for what constant b would the position of the particle at

!

t =1s be 2m?

2

4. A hollow hemisphere has center O and is fixed, with the plane of the rim is horizontal. A

particle A of mass m can move on the inside surface of the hemisphere. The particle is acted on by a horizontal force of magnitude P, whose line of action is in the vertical plane through O and A. Figure 3 shows the situation when A is in equilibrium, the line OA making and angle θ with the vertical.

a. If the inside surface of the hemisphere is smooth, find tan θ in terms of P, m and g. b. Suppose, instead that the inside surface is rough, with coefficient of static friction µs. The

particle is about to slip downwards. Show that:

!

tan" =P + µsmg

mg#µsP.

5. Maria and Guillermo climb to the top of the leaning tower of Pisa (height h = 50m). Maria

drops a cricket ball off the building, releasing it from rest; 1.0 seconds later Guillermo tosses a bocce ball (of the same diameter as the cricket ball, but twice as massive) off the building with a vertical speed v

0 downwards. The two balls pass one another halfway down the building.

Ignore air resistance in this problem. a. What was the speed v

0?

b. Are there any times at which the two balls have the same speed? Find the times or explain why there are no times.

c. Are there any times at which the two balls have the same acceleration? Find the times or explain why there are no times.

6. a) In the experiment “Dynamics of Rotational Motion”, a student called Pat has to determine the

radius of the hollow cylinder. Pat produces 11 measurements of its diameter with the help of a caliper whose vernier scale shows the reading error of 0.05 mm. The mean value of the diameter is calculated by adding all measurements and dividing the sum by the number of measurements. The calculator indicates the simple mean in centimeters as 9.2435225871. Pat decides to calculate the standard deviation for the diameter as if the measured values are distributed normally (according to the Gaussian distribution). For the value of standard deviation in centimeters, the calculator indicates 2.234579x10-3. What value for the radius of the cylinder (including error) should Pat write in the report? b) To determine acceleration due to gravity, a student measured time for a ball falling from a three-meter height. The measurements are given in the table.

Time, s 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 Number of outputs 1 2 9 8 14 11 6 3 2 1

Calculate the mean value for the time of the fall and the standard deviation for distribution of the values of time. Give a value (with error bars) for the acceleration due to gravity.

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Midterm: Aid Sheet f and g are functions of time (t) and A,B,c and n are constants, Derivative & Integral Formulas:

!

f (t) = c(t " B)n#

df

dt= cn(t " B)

n"1

!

c(t " B)n dtti

t f

# =c(t " B)n+1

n +1ti

t f

=c(t f " B)

n+1

n +1"c(ti " B)

n+1

n +1,(n $ "1)

!

f (t) = Acos(Bt)"df

dt= #ABsin(Bt)

!

f (t) + g(t)[ ]dt = f (t)dtti

t f

"ti

t f

" + g(t)dtti

t f

"

!

f (t) = Asin(Bt)"df

dt= ABcos(Bt)

!

d

dt( f + g) =

df

dt+dg

dt ,

!

d

dt( fg) =

df

dtg + f

dg

dt ,

!

d

dt[ f (g(t))]=

df

dg

dg

dt

Trigonometric Formulas:

!

sin2" + cos

2" =1

!

sin(" + #) = sin" cos# + sin# cos"

!

cos(" + #) = cos" cos# $ sin# sin"

Physics Formulas: (NOTE: you need to determine when these formulas apply, they are not necessarily always applicable)

!

r v =

dr r

dt

!

r a =

dr v

dt

!

r v (t f ) =

r v (ti) +

r a (t)dt

ti

t f

"

!

r r (t f ) =

r r (ti) +

r v (t)dt

ti

t f

"

!

r r =

r " r +

r v t

!

r u =

r u '+

r v

!

r a =

r " a

!

s = R"

!

" =d#

dt

!

" =2#

T

!

v = R"

!

" =d#

dt

!

r r (t) = xˆ i + yˆ j = Rcos "(t)[ ]ˆ i + Rsin "(t)[ ] ˆ j = Rˆ r

!

r v = "R# sin $(t)[ ]ˆ i + R# cos $(t)[ ] ˆ j = R# ˆ $

!

r a = a

rˆ r + a"

ˆ " = #R$ 2ˆ r + R

d$

dt

ˆ "

!

"(t f ) ="(ti) + #(t)dtti

t f

$

!

"(t f ) = "(ti) + #(t)dtti

t f

$

!

r F

net= m

r a

!

wapp =r n

!

Fk

= µk

r n

!

0 " Fs" µ

s

r n

Physical Constants:

!

g = 9.8m/s2

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Statistical Formulas:

N(x) = Ae

!(x! x )2

2" 2 , xest=

xi

i=1

N

!

N, !

est=

(i=1

N

" xi# x

est)2

N #1

!xest="

est

N, !"

est=

"est

2N # 2

66% of measurements are between x !1" and x +1! 95% of measurements are between x ! 2" and x + 2! 99% of measurements are between x ! 3" and x + 3! Propagation Formulas: Rule 1: If z = x + y or z = x ! y then: !z = (!x)

2+ (!y)

2

Rule 2: If z = x * y or z = x / y then: !zz=

!x

x

"#$

%&'

2

+!y

y

"

#$%

&'

2

Rule 3: If z = xn then: !z = nxn"1!x or equivalently: !z

z= n

!x

x