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Physics 5: Circular and Rotational Motion
A.
Circular Motion (5-1 to 5-3, 5-6 to 5-8)
1. constant perimeter (tangential) speed: vt = 2r/T
a. distance = circumference of the circle: 2r
b. time = time for one revolution: T (period)
2. constant inward (centripetal) acceleration: ac = v2/r
3. centripetal force, Fc = mac = mv2/r
a. turning on a road problems
v = 2r/T
ac


when the road is horizontal: Fc = Ff = smg
roads are banked in order to reduce the amount of
friction (component of the Fg is || to Fc)
b. horizontal loop problem (mass on a string)
Ft-x = Fc = mv2/r

Ft
Ft-y = Fg = mg
v = 2r/T





4.
Ft = (Fc2 + Fg2)½
tan = Fg/Fc ( is measured from horizontal)
c. vertical loop problem (mass on a string)
top: Fnet = Fc = Ft + Fg  Ft = Fc – Fg
Fg
Ft
Fg
Ft
bottom: Fnet = Fc = Ft – Fg  Ft = Fc + Fg
if on a roller coaster: Fn = Ft
Newton's law of universal gravity, Fg = GMm/r2
a. G = 6.67 x 10-11 N•m2/kg2
b. M = mplanet and m = msatellite
c. r is the distance, measured from center to center
d. g = GM/r2
e. Fg = Fc: GMm/r2 = mv2/r  v = (GM/r)½
v = 2r/T
m
Fg = Fg
M
r
Earth
Moon
Sun
Mass (kg)
5.98 x 1024
7.35 x 1022
1.99 x 1030
Radius (m)
6.38 x 106
1.74 x 106
6.96 x 108
r from Earth (m)
3.84 x 108
1.50 x 1011
Name __________________________
B.
Newton's Laws—Rotation (8-3 to 8-6)
1. torque,  = rFr (tau—Greek letter for t)
rotation
90o Fr
r
a.
2.
rcm
r = perpendicular distance from axis of rotation to
rotating force Fr
b. when r is not perpendicular to Fr, then  = rFrsin
c. torque units are m•N (not N•m—work)
First Law: Object remains at rest or uniform rotation as
long as no net torque (net) acts on it
a. measured as the moment of inertia, I = mr2
b.  corrects for mass distribution ( = 1 for a hoop)
c. equilibrium (net = 0)
1. center of mass for a complex system—balance
point where an upward torque generated by a
force equal to the total weight counters the
sum of all the downward weight torques
Fcm = (m1 + m2 + m3)g
m1
m2
m3
r1
r2
r3
F1
F2
F3
 not rotating  CM = 1 + 2 + 3
rcm(m1 + m2 + m3)g = r1m1g + r2m2g + r3m3g
rcm = (r1m1 + r2m2 + r3m3)/(m1 + m2 + m3)
rcm = (rimi)/mi
2. cantilever problems—how far from the edge
can m2 be placed without tipping?
mass of plank at its center
m1
m2
r1
r2
Fg1
not rotating 
Fg2
1 = 2
r1m1g = r2m2g
r1m1 = r2m2
3. two supports problems—what are the
tensions?
FL
mass of plank at its center
FR
rR
m1
m2
r1
r2
assume left side is point of rotation
not rotating 
R = 1 + 2
rRFR = r1m1g + r2m2g
FR = (r1m1g + r2m2g)/rR
solve for FL: FL + FR = m1g + m2g
3. Second Law: net = I (= rFr,  = mr2,  = a/r)
a. rFr = (mr2)(a/r)  Fr = ma (acceleration at rim)
b. pulley problems—what is the acceleration
m1
rough surface ()
m2
Fp – Ff = m1a + m2a

Fp – m1g = (m1 + m2)a
solve for d, v or t using kinematics
Fp
c. rolling problems—what is the acceleration

Frolling = ma + ma
m
Frolling = (1 +  )ma
Fgsin = (1 + )ma
solve for d, v or t using kinematics
Fgsin

C.
Conservation Laws—Rotation (8-7 to 8-8)
1. rotational momentum, L = rp = rmv (kg•m2/s)
a. when net = 0, then L = 0
b. change r and/or  will change v
orbiting planet
spinning diver
Kepler's Law
(A1  2 = A3  4)
r1v1 = r2v2
r11v1 = r22v2
2. rotational kinetic energy, Kr = ½mv2 (J)
a. v is the velocity at the rim
b. rolling kinetic energy: Krolling = ½(1 +  )mv2
3. mixed linear and rotation motion problems
a. Summary of translational and rotational formulas
Variable
Translational Rotational
Rolling
F = ma
force
Fr = ma
F = (1 + )ma
p = mv
momentum
L = rmv p + L = (1 + r)mv
kinetic energy K = ½mv2 Kr = ½mv2 K = ½(1 + )mv2
b. collision problems (conservation of momentum)—
what is the velocity after the boy jumps on the
merry-go-round?
m
v
convert linear into rotational (r = rM,  = 1)
Lm + LM = L'
M, rM, M
rMmmvm + rMMMvM = (rMmm + rMMM)v'
mvm = (mm + MM)v'
v' = mvm/(mm + MM)
c. pulley problems (conservation of energy)—what is
the velocity after descending h meters?
m1
rough surface ()
Ug3 – Wf1 = K1 + K2 + K3
m3gx – km1gx = ½(m1 + m2 + m3)v'2
v' = [2(m3 – km1)gx/(m1 + m2 + m3)]½
d.
m3

m2
pulley problems (conservation of energy)—what is
the velocity after descending h meters?
Ug1 + Ug2 = Ug1' + Ug2' + K1 + K2 + K3
0 + m2gh = m1gh + 0 + ½(m1 + m2 + m3)v'2
(m2 – m1)gh = ½(m1 + m2 + m3)v'2
v = [2(m2 – m1)gh/(m1 + m2 + m3)]½
m3

m2
m1
e.
h
rolling problems (conservation of energy)—what is
the velocity after descending h meters?
 m
Ug = Krolling

mgh = ½(1 + )mv2
h
gh = ½(1 + )v2
v = [2gh/(1 + )]½
f. cart problems (conservation of energy)—what is
the velocity after descending h meters?
m1
Ug = K + Krolling

 , m2
(m1 + 4m2)gh = ½m1v2 + ½(1 + )4m2v2
(m1 + 4m2)gh = ½(m1 + (1 + )4m2)v2
h

D.
Simple Harmonic Motion (SHM) (11-1 to 11-6)
1. oscillating mass on a spring
acceleration is NOT constant  kinematic
formulas are invalid
b. displacement, velocity and acceleration oscillate
between +A and –A, where A = amplitude
1. x = +A, when t = 0 (pictured above)
t=0
t = ¼T t = ½T t = ¾T
t=T
time
+A
-A
+A
displacement
0
0
-vmax
+vmax
velocity
0
0
0
-amax
+amax
-amax
acceleration
0
0
2. x = 0, when t = 0 (heading downward)
t=0
t = ¼T t = ½T t = ¾T
t=T
time
-A
+A
displacement
0
0
0
-vmax
+vmax
-vmax
velocity
0
0
+amax
-amax
acceleration
0
0
0
c. maximum acceleration, amax = A(k/m) = vmax2/A
Steps
Algebra
Fs = ma
start with
substitute kA for Fs
kA = ma
solve for a
amax = A(k/m)
d. maximum velocity, vmax = A(k/m)½ = 2A/T
Steps
Algebra
Us = K
start with
substitute ½kA2 for Us and ½mv2 for K ½kA2 = ½mv2
vmax = A(k/m)½
solve for v
e. time for one cycle, period, T = 2(m/k)½
Steps
Algebra
vmax = A(k/m)½
start with
substitute 2A/T for vmax
2A/T = A(k/m)½
simplify
2/T = (k/m)½
solve for T
T = 2/(k/m)½
substitute (m/k)½ for 1/(k/m)½
T = 2(m/k)½
f. formulas at midpoint, 0, and extremes, A
midpoint
extreme
x
xmax = A
0
v
0
vmax = A(k/m)½ = 2A/T
a
amax = -A(k/m) = vmax2/A
0
F
F = -ma = -kA
0
U
Umax = ½kA2
0
K
Kmax = ½mv2
0
a.
2.
pendulum
period of the simple pendulum, T = 2(L/g)½
Steps
Algebra
F = kx
start with
substitute mgsinrad for F
mgsinrad = kx
substitute Lrad for x
mgsinrad = kLrad
mg = kL
for small angles sinrad = rad
k = mg/L
solve for k
start with
T = 2(m/k)½
substitute mg/L for k
T = 2(m/mg/L)½
simplify
T = 2(L/g)½
b. notice that m cancels out of the equation, so the
period only depends on the L and g
3. damped harmonic motion
a.
a.
4.
amplitude of oscillating spring or swinging
pendulum will decrease until it stops—damping
b. damping is due to friction and air resistance
1. forces always oppose direction of velocity
2. damping is enhanced if oscillator is placed in
viscous fluid (car shock absorbers)
c. forced damping is accomplished with motors that
are programmed to oppose velocity (earthquake
protected buildings)
resonance
a. object can be set to oscillate by an external
force—forced vibration
b.
c.
when forced vibration matches natural vibration,
then amplitude builds with each vibration—
resonance
examples
1. child swinging
2. building during an earthquake
3. air inside a musical instrument
Experiments
1.
Centripetal Force Lab
a. Measure the time it takes the stopper to make 10
revolutions with the given lengths and hanging masses.
Control
string length, L
0.5 m
hanging weight, m1
200 g
stopper mass, m2
(3) Determine the percent difference between the
measured mr (true) and the calculated mr from (2).
b.
time (10 orbits), t
Double L
m1
string length, L
time (10 orbits), t
200 g
1.0 m
Half m1
m1
string length, L
time (10 orbits), t
100 g
0.5 m
Extend from the table edge a ½-m stick with 50-g at
40 cm, 10-g at 15 cm and 20-g at 5 cm and measure
the balance point (rcm).
50 g
10 g
20 g
mr
½-m stick|
|
|
|
|
table
50 40
25 15
5 0 cm
(1) Record the balance point rcm.
(2) Use the center of mass formula to determine rcm.
b.
Consider the free-body diagram.
Ftx

(3) Determine the percent difference between the
measured rcm (true) and the calculated rcm from (2).
Fty
Ft
c.
m1g
m1g
m2g
m2g
Ftx
Fty
What is Fty equal to?
Explore the relationship between center-of-mass and
balance by performing the following.
(1) Stand with your heels and back against a wall and
try to bend over and touch your toes. Explain
Which force equals Fc?
What is Ft equal to?
c. Calculate the following from the data.
Double L
Control
(2) Stand facing the wall with your toes against the
wall and try to stand on your toes. Explain
Half m1
T1 based on t
(3) Rest a meter stick on two fingers. Slowly bring
your fingers together. Explain
Ft based on m1
Fty based on m2
 based on Ft and Fty
Fc based on Ft and Fty
r based on L and 
v based on Fc
3.
1 +  lab
a. Roll the objects down an incline. Measure the change
in height, rolling distance and time it takes the object
to roll from the top to the bottom of the incline.
Ring
Disk
height, h
height, h
distance, d
distance, d
time, t
time, t
T2 based on v
% T1 and T2 (use
average as standard)
c. How was the period affected by the following?
Doubling L
2.
Halving m1
Equilibrium Lab
a. Extend from the table edge a ½-m stick with a 50-g
mass at 0 cm and measure the balance point (CM).
50 g
½-m stick
| rr  r50 
table |
|
|
50 cm
25 cm
0 cm
(1) Collect the following data.
r50
rr
ruler mass, mr
(2) Use the cantilever technique to determine the
mass of the ruler.
Ball
Cart
height, h
height, h
distance, d
distance, d
time, t
time, t
b.
Calculate the velocity for each using kinematics.
Ring
Disk
Ball
Cart
c.
Calculate (1 + ) for using conservation of energy.
Ring
Disk
Ball
Cart
d.
Calculate the percent difference with the ideal values.
Ring
Disk
Ball
Cart
2
1.5
1.4
1.0
4.
Simple Harmonic Motion Lab
a. Measure the length L and time t for 10 oscillations of a
spring with different hanging masses m, determine k
using two methods and compare the results.
(1) Collect the following data.
m (kg)
0
0.10
0.20
0.30
0.40
0.50
L (m)
t (s)
(2) Determine the following using length data.
added mass
0.10
0.20
0.30
0.40
Practice Problems
1.
2.
0.50
Fg Fg = mg
x x = L
3.
k F = kx
kaverage
(3) Determine the following using time data.
added mass
0.10
0.20
0.30
0.40
0.50
T T = t/10
4.
k T = 2(m/k)½
A. Circular Motion
When a tetherball is whirling around the pole, the net force
is directed
(A) toward the top of the pole
(B) toward the ground
(C) horizontally away from the pole
(D) horizontally toward the pole
You are standing in a bus that makes a sharp left turn.
Which of the following is true?
(A) you lean to the left because of centripetal force
(B) you lean to the right because of inertia
(C) you lean forward because of the net force is forward
(D) you lean to the right because of centrifugal force
You drive your car too fast around a curve and the car starts
to skid. What is the correct description of this situation?
(A) car's engine is not strong enough to keep the car from
being pushed out
(B) friction between the tires and the road is not strong
enough to keep the car in a circle
(C) car is too heavy to make the turn
(D) none of the above
A steel ball is whirling around in a circle on the end of a
string when the string breaks. Which path will it follow?
kaverage
(4) Calculate the percent difference between the two
values of k using the average as true.
b.
Measure the pendulum period for different releasing
angles  and see which angle gives the most ideal
values for T.
(1) Collect the following data.
L (m)
5o
10o
15o
20o
25o
(o)
T1 (s)
(2) Calculate the following from the data.
angle
5o
10o
15o
20o
x x = L(2/360)
F F = mgsin
25o
A B C
Two stones A and B have the same mass. They are tied to
strings and whirled in horizontal circles. The radius of the
circular path for stones A is twice the radius of stone B's
path. If the period of motion is the same for both stones,
what is the tension in cord A (FA) compared to cord B (FB)?
(A) FA = FB
(B) FA = 2FB
(C) FA = ½FB
6. You driving along a rural road. Which is true when you are
at the lowest point along a dip in the road?
(A) Fn < Fg
(B) Fn = Fg
(C) Fn > Fg
7. You swing a ball on the end of a string in a vertical circle.
Which is true of the centripetal force at the top of the circle?
(A) Fc = Ft + Fg (B) Fc = Ft – Fg (C) Fc = Fg – Ft
8. Which is stronger the Earth's pull on the Moon or the
Moon's pull on the Earth?
(A) Earth's pull (B) Moon's pull (C) they are equal
9. Is there a net force acting on an astronaut floating in orbit
around the Earth while on a space walk?
(A) yes
(B) no
10. A car is traveling clockwise on the north side of a circular
track. (r = 50 m) takes 16 s to make one lap. Determine
5.
v
k F = kx
direction of v
T T2 = 2(m/k)½
ac
%  T1 and T2
(3) Calculate T3 = 2(L/g)½.
direction of ac
11. The earth is 1.5 x 1011 m from the sun and makes one
complete circular orbit in 1 year.
a. What is the period of orbit in seconds?
(4) Which angle produced T closest to T3?
b.
What is the earth’s orbital velocity?
c.
What is the centripetal acceleration of the earth toward
the sun?
12. A driver of a 1000-kg sports car attempts a turn whose
radius of curvature is 50 m on a road where  = 0.8.
a. What is the fastest that the driver can make the turn?
b.
Could the driver make the turn at this speed
(1) with a 2,000-kg SUV? Explain
(2) when the road is wet? Explain
13. A 2-kg mass is moving at 5 m/s in a horizontal circle of
radius 1 m at the end of a cord.
a. What is the horizontal component of tension?
b.
What is the vertical component of tension?
c.
What is the overall tension in the cord?
d.
What is the tension in the cord at the bottom?
15. A 1-kg pendulum bob swings back and forth from a 2-m
string that can support 15 N of tension without breaking.
a. What is the maximum speed that the bob can reach at
the bottom of the swing without breaking the string?
b.
20. Consider the following changes to earth.
I
Increase earth's mass
II Decrease earth's mass
III Increase earth's radius IV Decrease earth's radius
Which changes would decrease the acceleration
due to gravity on the earth's surface?
Which changes would increase the acceleration
due to gravity on the earth's surface?
Which changes would decrease the acceleration
due to gravity on the moon?
Which changes would increase the acceleration
due to gravity on the moon?
21. The Earth's mass is 81 times the mass of the moon and the
Earth's radius is 4 times the radius of the moon.
a. What is gMoon in terms of gEarth?
b.
What is the mass of a 50 kg person on the Moon?
c.
What is the weight of a 50 kg person on the Moon?
What angle does the cord make with the horizontal?
14. A 2-kg mass is moving at 5 m/s in a vertical circle of radius
1 m at the end of a cord.
a. What is the tension in the cord at the top of the circle?
b.
19. When the Apollo Missions went to the moon they passed a
point where the gravitational attractions from the moon and
the earth are equal. What is the ratio rEarth/rMoon where this
happened if mEarth/mMoon = 100?
What is the maximum height measured from vertical
that the bob can reach?
B. Newton's Laws—Rotation
22. You are using a wrench to loosen a rusty nut. Which will
produce the greatest torque?
A
B
C
D
23. 3 identical balls descend 3 identical ramps (except for s).
Ball A slides down ramp A (s = 0), ball B rolls down ramp B
(s = .3) and ball C rolls down ramp C (s = .6). Which is
true of their velocities when the reach the end of their ramp?
(A) vA > vB = vC (B) vA > vB > vC (C) vA = vB = vC
Questions 24-25 Four objects have the same mass and radius.
F
axis of rotation  
16. How would the force of gravity be affected if the Earth
a. had the same mass but a smaller radius?
Between Earth and Moon
On the Earth's surface
b. had the same radius but a smaller mass?
Between Earth and Moon
On the Earth's surface
17. Determine the acceleration due to gravity on the planet
compared to Earth.
Mass
Radius (x Earth)
Acceleration (x gEarth)
m = mEarth
r = rEarth
g
m = mEarth
r = 2rEarth
m = mEarth
r = ½rEarth
m = 2mEarth
r = rEarth
m = ½mEarth
r = rEarth
18. What is the acceleration due to gravity (g) on Mars?
( m = 6.4 x 1023 kg, r = 3.4 x 106 m)
(A) hollow cylinder,  = 1 (B) solid cylinder,  = 1/2
(C) hollow ball,  = 2/3
(D) solid ball,  = 2/5
24. Which would have the greatest moment of inertia?
25. Which would have the greatest rotational acceleration?
26. A 1-kg block is hung at the end of a rod 1-m long. The
balance point is 0.25 m from the end holding the block, what
is the mass of the rod?
| 0.25 m | 0.25 m |
center of rod
1 kg
(A) 0.25 kg (B) 0.5 kg (C) 1 kg
(D) 2 kg
27. What is the total mass of the mobile? (rods are massless)
1m
B
(A) 5 kg
1m
(B) 6 kg
2m
3m
A
1 kg
(C) 7 kg
(D) 8 kg
28. Consider the two configurations of interlocking blocks on the
edge of a table. Which of the following is true?
A
B
(A) A tips (B) B tips (C) both tip (D) neither tip
29. Consider the door as viewed from above.
F1
F2 
34. A plank is placed on two scales, which are then zeroed. A
172-cm-tall student lies on the plank resulting in the
reading shown.
Determine
a. The torque when F1 = 45 N and r1 = 1 m.
b.
The force, F2, where r2 = 0.4 m, that will generate the
same torque as part a.
30. A 5-kg disk ( = ½) rolls down a 30o incline. Determine
a. The parallel component of Fg.
a.
What is the student's mass?
b.
What is the distance from her feet to her center-of-mass?
35. A 2200-kg trailer is attached to a stationary truck.

b.
The disk's acceleration at the rim.
31. A 25-kg box rests on the edge of a merry-go-round (r = 2 m).
a. What is the maximum force of friction between the box
and merry-go-round (s = 0.80)?

b.
Determine the normal force at A and B
What is the maximum velocity before the box slips off?

c.
What is the acceleration of the 200-kg merry-go-round
( = ½) exerting by a 50-N force along the outer rim?

d.
A
B

36. A 200-N sign hangs from the end of a 5-m pole, which is
held at a 37o angle by a horizontal guy wire.
guy wire
How much time will it take to reach the maximum
velocity before the box slips off of the merry-go-round?

e.
pole
Would this time increase or decrease if  = 1.0?

32. A 5-m, 75-kg plank is extended 2 m over the edge of a
building. What is the maximum distance that a 25-kg child
walks out from the building's edge without tipping the
plank?
37o
Physics
is Phun
Determine the tension in the guy wire.
C. Conservation Laws—Rotation
37. A hoop, cylinder and sphere roll down a 1-m ramp inclined
30o at the same time that a box slides down a frictionless
ramp that is also 1 m long and inclined 30o.
a. Derive a formula for determining the velocity of each
object when it reaches the bottom of the ramp.

33. Consider the diagram of the printing press on a table.
b.
What are the velocities when they reach the bottom?
Hoop ( = 1)
Cylinder ( = 1/2)
Sphere ( = 2/5)
Box ( = 0)
c.
Determine
What is the order in which they reach the bottom?
38. Determine the velocity of a Yo-Yo ( = ½) that "rolls"
straight down its string a distance of 0.50 m.
43. What is the angular momentum of a 0.2-kg ball traveling at
9 m/s on the end of a string in a circle of radius 1 m?
39. A marble ( = 2/5) rolls from rest down a ramp and around
a loop (radius = 10 m). Determine
A
44. What is the angular momentum of Earth, m = 6.0 x 1024 kg?
a. about its axis of rotation ( = 2/5, rplanet = 6.4 x 106 m)
B
H
10 m
a.
the minimum velocity at B.
b.
the minimum height H at A.
b.
in its orbit around the Sun ( = 1, rorbit = 1.5 x 1011 m)
45. Halley's comet follows an elliptical orbit, where its closest
approach to the sun is 8.9 x 1010 m and its farthest
distance is 5.3 x 1012 m. How many times faster does the
comet travel at its fastest compared to its slowest?
40. A string is attached to a 1.0-kg block and is wrapped round
a pulley ( = ½, m = 2.0 kg). The block is released from
rest and accelerates downward while the pulley rotates.
What is the block's velocity after descending 1 m?
41. Two weights (m1 = 0.60 kg, m2 = 0.40 kg) are connected
by a cord that hangs from a pulley ( = ½, M = 0.50 kg).
46. A child (m = 42 kg) runs toward a stationary merry-go-round
( = ½, m = 180 kg, r = 1.2 m) along a tangent at 3 m/s. The
child jumps on the merry-go-round and sets it rotating.
3 m/s
42 kg
=½
180 kg
1.2 m
What is the speed of the merry-go-round after the child
jumps on?
47. The rim of a disk ( = ½, m = M, r = R) rotates at a velocity,
V. A ring ( = 1, m = M, r = R) is dropped on top of the disk.
a. Calculate Ltotal before the ring is dropped on the disk.
M
b.
Calculate the velocity after the ring is dropped.
m1
1m
m2
What is the velocity of m1 after descending 1 m?
48. Tarzan (100 kg) is on a ledge that is 20 m above Jane (45
kg), who is trapped on a lower ledge. Tarzan grabs a long
vine and swings down from the ledge and grabs Jane, who
is stationary. The two swing over to a rock ledge on the
other side of the river gorge that is 10 m higher than the
rock ledge where Jane is trapped. Assuming the vine is
long enough, can Tarzan and Jane reach the other side?
42. A string attached to a 20-kg block resting on a table
passes over a pulley ( = ½, m = 4 kg) and attaches to a
14-kg mass hanging over the edge of the table. The 20-kg
box slide along the table ( = 0.25) while the 14-kg mass
descends 1 m.
T
J
20 kg
a.
Calculate Tarzan's velocity when he grabs Jane.
b.
Calculate the velocity after Tarzan grabs Jane.
c.
Calculate how high Tarzan swings to the other side.
1m
14 kg
What is the hanging mass' velocity after descending 1 m?
What is the velocity of the student after he jumps on to the
merry-go-round?
d.
e.
Did Tarzan and Jane make it?
What could Tarzan have done to save Jane?
f.
How high would Tarzan have to start to save Jane?
g.
What minimum initial velocity would Tarzan need to
save Jane starting from the original ledge?
49. A 1-kg, disk ( = ½) is placed on a 2-m ramp where the top
is 1 m above the base of the ramp. The disk is placed at the
top and rolls down to the base of the ramp.
a. What is the disk's velocity when reaches the base?
b.
How much time does it take the disk to travel the 2 m?
c.
Predict how the following alterations would change the
disk's velocity at and time to reach the base of ramp?
Alteration
Final Velocity
Time
A 2.0-kg disk is used
A 1.0-kg ring ( = 1) is used
A 3-m ramp is used, but h = 1 m
50. A string attached to a 10-kg box resting on a table passes
over a pulley ( = ½, m = 1 kg) and attaches to a 5-kg mass
hanging over the edge of the table. The 10-kg box slide 1 m
along the table ( = 0.3) while the 5-kg mass descends.

54. A 2-kg block and a 1-kg sphere hang from 2-m strings. The
sphere is raised to a horizontal position and swings toward
the block and collides with it.
1 kg
a.
2 kg
What is the sphere's velocity before the collision?
Assume that the collision is inelastic.
b. What is the sphere-block's velocity after the collision?
c.
What is the maximum height reached after the collision?
d.
What is the maximum height reached after the collision
if the block and sphere exchange positions initially?
The sphere is raised to a horizontal position initially and
then collides elastically with the block.
e. What are the velocities after the collision?
1m
a.
How much kinetic energy does the system have at the
point where the 5-kg mass has descended 1 m?
b.
What is the maximum velocity of the system?

104
51. Halley's Comet has a velocity of 3.88 x
m/s when it is
8.9 x 1010 m from the sun. How fast is it traveling when it is
5.3 x 1012 m from the sun?
52. What is the angular momentum of the Moon?
(m = 7.35 x 1022 kg, rmoon = 1.74 x 106 m, rorbit = 3.84 x 108 m,
Torbit = Trotation = 2.42 x 106 s)
a. about its axis of rotation ( = 2/5)
b.
in its orbit around the Earth ( = 1)
53. A student (m = 75 kg) runs at 5 m/s tangentially toward a
merry-go-round ( = ½, m = 150 kg, r = 2 m) rotating at 2
m/s, jumps on the merry-go-round and sets it rotating.
f.
What are the maximum heights reached by the block
and sphere?
g.
Was potential energy conserved after the collision?
D. Simple Harmonic Motion (SHM)
Questions 55-58 A spring bob in SHM has amplitude A and
period T.
55. What is the total distance traveled by the bob after time T?
(A) 0
(B) ½A
(C) 2A
(D) 4A
56. What is the total displacement after time T?
(A) 0
(B) ½A
(C) 2A
(D) 4A
57. How long does it take the bob to travel a distance of 6A?
(A) ½T
(B) ¾T
(C) 5/4T
(D) 3/2T
58. At what point in the motion is v = 0 and a = 0 simultaneously?
(A) x = 0
(B) 0 < x < A(C) x = A
(D) no point
59. A mass on the end of a spring oscillates in simple harmonic
motion with amplitude A. If the mass doubles but the
amplitude is not changed, what happen to the total energy?
(A) decrease
(B) no change (C) increase
60. If the amplitude of a simple harmonic oscillator is doubled,
which quantity will change the most?
(A) T
(B) v
(C) a
(D) K + U
61. A spring with mass m has period T. If m is doubled, what is
the new T?
(A) T/2
(B) T
(C) 2T
(D) 2T
Questions 62-63 Consider the periods of pendulums A and B,
62. Which period is greater when LA = LB, but mA > mB?
(A) A
(B) B
(C) tie
63. Which period is greater when mA = mB, but LA > LB?
(A) A
(B) B
(C) tie
64. A grandfather clock has a weight at the bottom of the
pendulum that can be moved up or down. If the clock is
running slow, should the weight be moved up or down?
(A) up
(B) down
(C) neither will work
65-66 Consider the following options.
(A) add mass to the oscillator
(B) move the oscillator to an elevator accelerating down
(C) move the oscillator to an elevator accelerating up
(D) move the oscillator to the Moon
65. Which will decrease the period of a pendulum?
66. Which will change the period of oscillation on a spring?
67. After a pendulum starts swinging, its amplitude gradually
decreases with time because of friction. What happens to
the period of the pendulum during this time
(A) decreases (B) no change (C) increases
68. When you sit on a swing, the period of oscillation is T1.
When you stand on the same swing, the period of
oscillation is T2. Which is true?
(A) T1 < T2
(B) T1 = T2
(C) T1 > T2
69. When a 50 kg person sits on a swing, the period of
oscillation is T1, when a 100 kg person sits on the same
swing, the period of oscillation is T2. Which is true?
(A) T1 < T2
(B) T1 = T2
(C) T1 > T2
70. Consider the graph
of one cycle of SHM.
a.
Determine the time in terms of T for each situation.
Maximum up
Zero
Maximum down
Acceleration
Velocity
b. Determine the following when m = 1 kg, k = 100 N/m
and A = 0.1 m.
amax
vmax
T
Kmax
Umax
c.
Graph the potential energy (----), kinetic energy (•••)
and total energy (––) for one complete oscillation.
0.5 J
0J
¼T
¾T
complete the following chart (x = +A at t = 0 s)
t
¼T
½T
¾T
1T
x
d.
v
a
F
e. How do the following change if the amplitude is 0.2 m?
Max acceleration
Max velocity
Period
71. A 1-kg ball on the end of a 1-m string is set in motion by
pulling the ball out so that it is raised 0.015 m. Determine
a. the maximum speed
b.
the period of oscillation.
c.
What would the period be with the following changes?
m = 4 kg
L=4m
g = 40 m/s2
Practice Multiple Choice
Briefly explain why the answer is correct in the space provided.
1. In the diagram, a car travels clockwise at constant speed.
Which letters represent the directions of the car's velocity,
v, and acceleration, a?
v
a
v
a
(A) A
C
(B) C
B
(C) C
A
(D) D
A
72. Consider the diagram of one cycle of SHM.
2.
A racing car is moving around the circular track of radius
300 m. At the instant when the car's velocity is directed
due east, its acceleration is 3 m/s2 directed due south.
When viewed from above, the car is moving
(A) clockwise at 30 m/s (B) counterclockwise at 30 m/s
(C) clockwise at 10 m/s d. counterclockwise at 10 m/s
3.
A person weighing 800 N on earth travels to another planet
with the same mass as earth, but twice the radius. The
person's weight on this other planet is most nearly
(A) 200 N (B) 400 N (C) 800 N (D) 1600 N
4.
The disk is rotating counterclockwise when the ball is
projected outward at the instant the disk is in the position
shown.
a.
Determine the time (in terms of T) for each situation.
Maximum up
Zero
Maximum down
Acceleration
Velocity
b. Determine the following when m = 1 kg, k = 100 N/m
and A = 0.25 m.
amax
vmax
T
Kmax
Umax
c.
Graph the potential energy, kinetic energy and total
energy for one complete oscillation.
Which of the following best shows the subsequent direction
of the ball relative to the ground?
(A) 
(B) 
(C) 
(D) 
3J
0J
¼T
¾T
complete the following chart (x = 0 at t = 0 s)
t
¼T
½T
¾T
x
5.
d.
1T
A ball is released from rest at position P swings through
position Q then to position R where the string is again
horizontal.
v
a
F
e. Which is the same when the amplitude is increased to
0.50 m?
maximum acceleration maximum velocity
period
73. A 1-kg ball swings from the ceiling on the end of a 2-m
string. The ball starts its swing from a position that is
0.2 m above its lowest point.
a. What is the maximum speed of the ball?
b.
What is the period of oscillation for the pendulum?
What are the directions of the ball's acceleration at
positions, Q and R?
Q
R
Q
R
(A) 

(B) 

(C) 

(D) 

6.
A 5-kg sphere is connected to a 10-kg sphere by a rod.
The center of mass is closest to
(A) A
(B) B
(C) C
(D) D
7.
A ball attached to a string is moved at constant speed in a
horizontal circular path. A target is located near the path of
the ball as shown in the diagram.
13. A square piece of plywood on a horizontal tabletop is
subjected to the two horizontal forces shown above.
Where should a third force of magnitude 5 N be applied to
put the piece of plywood into equilibrium?
A B
C
At which point along the ball's path should the string be
released, if the ball is to hit the target?
(A) A
(B) B
(C) C
(D) D
D
Questions 8-9 refer to a ball that is tossed straight up from the
surface of a small asteroid with no atmosphere. The ball
rises to a height equal to the asteroid's radius and then
falls straight down toward the surface of the asteroid.
8. What forces act on the ball while it is on the way up?
(A) a decreasing gravitational force that acts downward
(B) an increasing gravitational force that acts downward
(C) a constant gravitational force that acts downward
(D) a constant gravitational force that acts downward and
a decreasing force that acts upward
14. A 5-m, 100-kg plank rests on a ledge with 2 m extended out.
9.
15. The radius of the earth is approximately 6,000 km. The
acceleration of an astronaut in a perfectly circular orbit
6,000 km above the earth would be most nearly
(A) 0 m/s2 (B) 2.5 m/s2 (C) 5 m/s2 (D) 10 m/s2
The acceleration of the ball at the top of its path is
(A) at its maximum value for the ball's flight
(B) equal to the acceleration at the surface
(C) equal to one-half the acceleration at the surface
(D) equal to one-fourth the acceleration at the surface
10. The diagram shows a 5.0-kg bucket of water being swung
in a horizontal circle of 0.70-m radius at a constant speed
of 2.0 m/s.
The centripetal force on the bucket of water is
(A) 5.7 N
(B) 29 N
(C) 14 N
(D) 200 N
Questions 11-12 A 125-N board is 4 m long and is supported by
vertical chains at each end. A person weighing 500 N is
sitting on the board. The tension in the right chain is 250 N.
11. What is the tension in the left chain?
(A) 250 N (B) 375 N
(C) 500 N (D) 625 N
12. How far from the left end of the board is the person sitting?
(A) 0.4 m (B) 1.5 m
(C) 2 m
(D) 2.5 m
How far can a 50-kg person walk out on the plank past the
edge of the building before the plank just begins to tip?
(A) ½ m
(B) 1 m
(C) 3/2 m
(D) 2 m
16. The diagram represents two satellites of equal mass, A
and B, in circular orbits around a planet.
Comparing the gravitational force between satellite and
planet, B's gravitational force compared to A's is
(A) half as great
(B) twice as great
(C) one-fourth as great
(D) four times as great
17. The system is balanced when hanging by the rope.
What is the mass of the fish?
(A) 1.5 kg (B) 2 kg
(C) 3 kg
(D) 6 kg
18. A ball attached to a string is whirled around in a horizontal
circle with radius r, speed v and tension T. If the radius is
increased to 4r and the tension remains the same, then the
speed of the ball is
(A) ¼v
(B) ½v
(C) v
(D) 2v
19. A 0.4-kg object swings on the end of a string. At the
bottom of the swing, the tension in the string is 6 N. What
is the centripetal force acting on the object at the bottom of
the swing?
(A) 2 N
(B) 4 N
(C) 6 N
(D) 10 N
20. Two wheels, fixed to each other, are free to rotate about a
frictionless axis perpendicular to the page. Four forces are
exerted tangentially to the rims of the wheels.
The net torque on the system about the axis is
(A) zero
(B) FR
(C) 2FR
(D) 5FR
21. Mars has a mass 1/10 that of Earth and a diameter 1/2 that
of Earth. The acceleration of a falling body near the
surface of Mars is most nearly
(A) g/5
(B) 2g/5
(C) g/2
(D) g
22. A satellite of mass m and speed v moves in a stable,
circular orbit around a planet of mass M. What is the radius
of the satellite’s orbit?
(A) GM/mv (B) Gv/mM (C) GM/v2 (D) GmM/v
25. An asteroid moves in an elliptic orbit with the Sun at one
focus.
Which of the following increases as the asteroid moves
from point P in its orbit to point Q?
(A) Speed
(B) Angular momentum
(C) Total energy
(D) Potential energy

26. A satellite S is in an elliptical orbit around a planet P with r1
and r2 being its closest and farthest distances, respectively,
from the center of the planet. If the satellite has a speed v1
at its closest distance, what is its speed at its farthest
distance?
(A) (r1/r2)v1
(B) (r2/r1)v1
(C) (r2 – r1)v1
(D) ½(r1 + r2)v1
27. A satellite of mass M moves in a circular orbit of radius R
at a constant speed v. Which must be true?
I. The net force on the satellite is equal to Mv2/R
and is directed toward the center of the orbit.
II. The net work done on the satellite by gravity in
one revolution is zero.
III. The angular momentum of the satellite is a
constant.
(A) I only
(B) III only (C) I and II (D) I, II, and III
28. For which motions is there a variable force involved?
(A) Constant speed in a straight line
(B) Simple harmonic motion
(C) Constant speed in a circle
(D) Constant acceleration in a straight line
Questions 29-30 A sphere of mass M, radius R, and = 2/5, is
released from rest at the top of an inclined plane of height h.
23. A wheel of radius R is mounted on an axle so that the wheel
is in a vertical plane. Three small objects having masses m,
M, and 2M, respectively, are mounted on the rim.
29. If the plane is frictionless, what is the speed of the center
of mass of the sphere at the bottom of the incline?
(A) (2gh)½ (B) 2Mgh
(C) 2MghR2 (D) 5gh
What is m in terms of M when the wheel is stationary?
(A) 1/2 M
(B) M
(C) 3/2 M
(D) 2 M

30. If the plane has friction so that the sphere rolls without
slipping, what is the speed at the bottom of the incline?
(A) (2gh)½ (B) 2Mgh
(C) 2MghR2 (D) (10gh/7)½
24. In each case the unknown mass m is balanced by a known
mass M1 or M2.
31. A particle of mass, m, moves with a constant speed v
along the dashed line y = a.
What is the value of m in terms of the known masses?
(A) M1 + M2
(B) (M1 + M2)/2
(C) M1M2
(D) (M1M2)½

When the x-coordinate of the particle is xo, the magnitude
of the angular momentum of the particle with respect to the
origin of the system is
(A) zero
(B) amv
(C) xomv
(D) (vo2 + a2)½mv
40. The graph is of the displacement x versus time t for a
particle in simple harmonic motion with a period of 4 s.
Questions 32-36 A block oscillates without friction on the end of
a spring. The minimum and maximum lengths of the
spring as it oscillates are, respectively, xmin and xmax.
The graphs below can represent quantities associated with
the oscillation as functions of the length x of the spring.
(A)
(B)
(C)
(D)
Which graph shows the potential energy of the particle as
a function of time t for one cycle of motion?
(A)
(B)
(C)
(D)
41. Two identical springs are hung from a horizontal support.
When a 1.2-kg block is suspended from the pair of
springs, each spring is stretched an additional 0.15 m.
32. Which graph can best represent the total mechanical
energy of the block-spring system as a function of x?
The spring constant of each spring is most nearly
(A) 40 N/m (B) 48 N/m (C) 60 N/m (D) 80 N/m
33. Which graph can best represent the kinetic energy of the
block as a function of x?
34. Which graph can best represent the potential energy of the
block as a function of x?
35. Which graph can best represent the acceleration of the
block as a function of x?
36. Which graph can best represent the velocity of the block as
a function of x?
37. A block attached to the lower end of a vertical spring
oscillates up and down. The period of oscillation depends
on which of the following?
I. Mass of the block
II. Amplitude of the oscillation
III. Spring constant
(A) I only
(B) II only (C) III only (D) I and III only
38. When a 1-kg bob is attached to a spring, the period of
oscillation is 2 s. What is the period of oscillation when a
2-kg bob is attached to the same spring?
(A) 0.5 s
(B) 1.0 s
(C) 1.4 s
(D) 2.8 s
39. A pendulum and a mass hanging on a spring both have a
period of 1 s on Earth. They are taken to planet X, which
has twice the gravitational acceleration g as Earth. Which
is true about the periods of the two objects on planet X
compared to their periods on Earth?
(A) Both are shorter.
(B) Both are longer.
(C) The pendulum is longer and the spring is the same.
(D) The pendulum is shorter and the spring is the same.
42. A ball is dropped from a height of 10 m onto a hard surface
so that the collision at the surface may be assumed elastic.
Under such conditions the motion of the ball is
(A) simple harmonic with a period of about 1.4 s
(B) simple harmonic with a period of about 2.8 s
(C) simple harmonic with an amplitude of 5 m
(D) periodic but not simple harmonic
43. An object swings on the end of a cord as a simple
pendulum with period T. Another object oscillates up and
down on the end of a vertical spring, also with period T. If
the masses of both objects are doubled, what are the new
values for the periods?
Pendulum Spring
Pendulum Spring
(A) T/√2
√2T
(B) T
√2T
(C) T
T
(D) √2T
T
44. When a mass m is hung on a spring, the spring stretches a
distance d. If the mass is then set oscillating on the spring,
the period of oscillation is proportional to
(A) (d/g)½ (B) (g/d)½ (C) (d/mg)½ (D) (m2g/d)½
45. A 3-kg block is hung from a spring, causing it to stretch 12
cm at equilibrium. The 3-kg block is then replaced by a 4kg block, and the new block is released from the spring
when it is unstretched. How far will the 4-kg block fall
before its direction is reversed?
(A) 9 cm
(B) 18 cm
(C) 24 cm
(D) 32 cm
46. A spring is fixed to the wall at one end. A block of mass M
attached to the other end of the spring oscillates with
amplitude A on a frictionless, horizontal surface. The
maximum speed of the block is v.
The spring constant is
(A) Mg/A
(B) Mgv/2A (C) Mv2/2A (D) M(v/A)2
47. A sphere of mass m1 is attached to a spring. A second
sphere of mass m2 is suspended from a string of length L,
If both spheres have the same period of oscillation, which
of the following is an expression for the spring constant?
(A) L/m1g (B) g/m2L (C) m1L/g (D) m1g/L
Practice Free Response
1.
A roller coaster ride at an amusement park lifts a car of
mass 700 kg to point A at a height of 90 m above the
lowest point on the track, as shown above. The car starts
from rest at point A, rolls with negligible friction down the
incline and follows the track around a loop of radius 20 m.
Point B, the highest point on the loop, is at a height of 50 m
above the lowest point on the track.
a.
3.
P
(1) Indicate on the figure the point P at which the
maximum speed of the car is attained.
(2) Calculate the value vmax of this maximum speed.
b.
Calculate the speed vB of the car at point B.
c.
(1) Draw and label vectors to represent the forces
acting on the car when it is upside down at point B.
b.
force of gravity on the 10 kg mass.
c.
net force rotating the pulley.
d.
acceleration of the pulley at the rim.
e.
velocity when the system has moved 2 m.
f.
loss of potential energy as the 10-kg mass falls 2 m.
g.
work done by friction as the 20-kg block slides 2 m.
h.
velocity of the system
A 0.5-kg hoop ( = 1) rolls from rest at the top of the ramp of
length L = 2 m and angle  = 30o. The table height H = 1 m.
a.
Determine the potential energy of the hoop at the top
of the ramp, where Ug = 0 at the floor.
b.
The hoop rolls down the ramp and then onto the floor.
Determine the hoop's
(1) speed at the bottom of the ramp.
(2) speed just before it hits the floor.
(3) translational kinetic energy before it hits the floor.
(4) percentage of total energy that is rotational kinetic
energy just before it hits the floor.
(2) Calculate all the forces identified in (c1).
2.
c.
A string attached to a 20-kg block resting on a table
passes over a pulley ( = ½, m = 10 kg) and attaches to a
10-kg mass hanging over the edge of the table. The 20-kg
box slide along the table ( = 0.30) while the 10-kg mass
descends 2 m.
The hoop is replaced by a disk ( = ½). Comparing the
hoop ( = 1) with the disk ( = ½) just before they
lands on the floor, which would
(1) have the greatest % rotational kinetic energy?
(2) land furthest from the base of the table?
(3) have the most kinetic energy just before it landed?
2m
Determine the
a. force of friction on the 20-kg block as it slides.
4.
The graph shows a system in simple harmonic motion.
Complete the chart with either +, 0, or –.
t
0s
1s
2s
3s
x (m)
v (m/s)
a (m/s2)
F (N)
U (J)
K (J)
5. A 3.0 kg bob swings on the end of a 1.0 m string. The
potential energy U of the object as a function of distance x
from its equilibrium position is shown. This particular
object has a total energy E of 0.4 J.
6.
a.
What is the greatest distance x for the pendulum bob?
b.
How much time does it take the pendulum to go from
the greatest +x to the greatest –x?
c.
Determine the bob's kinetic energy when x = -7 cm.
d.
What is the object's speed at x = 0?
The cart of mass m with four wheels each of mass m/4 and
 = ½ is released from rest and rolls from height h. After
rolling down the ramp and across the horizontal surface,
the cart collides and sticks with a bumper of mass 3m
attached to a spring, which has a spring constant k.
Given: m = 1 kg, h = 0.50 m, k = 250 N/m, determine
a. the potential energy of the cart at the top of the ramp.
b.
the speed of the cart at the bottom of the ramp.
c.
the carts translational kinetic energy before the collision.
d.
the speed of the cart just after the collision.
e.
the translational kinetic energy of the cart and bumper
just after the collision.
f.
the amount that the spring is compressed.