Physics 5: Circular and Rotational MotionName ______
A.Circular Motion (5-1 to 5-3, 5-6 to 5-8)
1.constant perimeter (tangential) speed: vt = 2r/T
a.distance = circumference of the circle: 2r
b.time = time for one revolution: T (period)
2.constant inward (centripetal) acceleration: ac = v2/r
3.centripetal force, Fc = mac = mv2/r
The key to solving circular motion problems is to ask: what is causing the object to move in a circle?
a.turning on a road problems
v = 2r/Tac
- 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 = 2r/T
- Ft = (Fc2 + Fg2)½
- tan = Fg/Fc ( is measured from horizontal)
c.vertical loop problem (mass on a string)
- top: Fnet = Fc = Ft + FgFt = Fc – Fg
Fg Ft
Fg Ft
- bottom:Fnet = Fc = Ft – FgFt = Fc + Fg
- if on a roller coaster: Fn = Ft
4.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 = 2r/T
m Fg = Fg M
r
Mass (kg) / Radius (m) / r from Earth (m)Earth / 5.98 x 1024 / 6.38 x 106
Moon / 7.35 x 1022 / 1.74 x 106 / 3.84 x 108
Sun / 1.99 x 1030 / 6.96 x 108 / 1.50 x 1011
Practice Problems
A.Circular Motion
1.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
2.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
3.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
4.A steel ball is whirling around in a circle on the end of a string when the string breaks. Which path will it follow?
A B C
5.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)FnFg(B)Fn = Fg(C)FnFg
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
vdirection ofv
ac
direction ofac
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?
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 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.What is the tension in the cord at the bottom?
15.A1-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.What is the maximum heightmeasured from vertical that the bob can reach?
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 surfaceb.had the same radius but a smaller mass?
Between Earth and Moon / On the Earth's surface17.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)
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?
20.Consider the following changes to Earth.
IIncrease Earth's massIIDecrease Earth's mass
IIIIncrease Earth's radiusIVDecrease Earth's radius
Which changes would decrease the acceleration due to Earth's gravity on the Earth's surface?Which changes would increase the acceleration due to Earth's gravity on the Earth's surface?
Which changes would decrease the acceleration due to Earth's gravity on a satellite?
Which changes would increase the acceleration due to Earth's gravity on a satellite?
21.The Earth's mass is about 80 times the mass of the Moon and the Earth's radius is about 4 times the radius of the Moon.
a.What is gMoonin 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?
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)vAvB = vC(B)vAvBvC(C)vA = vB = vC
Questions 24-25 Four objects have the same mass and radius.
F
axis of rotation
(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)
1 m 2 m A
B 1 m 3 m 1 kg
(A)5 kg(B)6 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.
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)rollsdown a 30o incline. Determine
a.The parallel component of Fg.
b.The disk's acceleration at the rim.
31.A 25-kg box rests on the edge of amerry-go-round (r = 2 m).
a.What is the maximum force of friction between the box and merry-go-round (s = 0.80)?
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.How much time will it take to reach the maximum velocity before the box slips off of the merry-go-round?
e.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 a25-kg child walks out from the building'sedge without tipping the plank?
33.Consider the diagram of the printing press on a table.
Determine
F1F2
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.
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.
Determine the normal force at A and B.
AB
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
pole
37o
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.
b.What are the velocities when they reach the bottom?
Hoop ( = 1)Cylinder ( = 1/2)
Sphere ( = 2/5)
Box( = 0)
c.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.
39.A marble ( = 2/5) rolls from rest down a ramp and around a loop (radius = 10 m). Determine
A
B
H 10 m
a.the minimum velocity at B.
b.the minimum height H at A.
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).
M
m1
1 m
m2
What is the velocity ofm1after descending 1 m?
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.
20 kg
1 m
14 kg
What is the hanging mass' velocity after descending 1 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?
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.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?
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.
b.Calculate the velocity after the ring is dropped.
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?
T
J
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.
d.Did Tarzan and Jane make it?e.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 / TimeA 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.
1 m
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?
51.Halley's Comet has a velocity of 3.88 x 104 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.What is the velocity of the student after he jumps on to the merry-go-round?
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
2 kg
a.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?
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 mAmB?
(A)A(B)B(C)tie
63.Which period is greater when mA = mB, but LALB?
(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 downAcceleration
Velocity
b.Determine the following when m = 1 kg, k = 100 N/m and A = 0.1 m.
amaxvmax
T
Kmax
Umax
c.Graph the potential energy (----), kinetic energy (•••) and total energy (––) for one complete oscillation.
0.5 J0 J
¼T / ¾T
d.complete the following chart (x = +A at t = 0 s)
t / ¼T / ½T / ¾T / 1Tx
v
a
F
e.How do the following changeif the amplitude is 0.2 m?
Max acceleration / Max velocity / Period71.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 = 4 m / g = 40 m/s272.Consider the diagram of one cycle of SHM.
a.Determine the time (in terms of T) for each situation.
Maximum up / Zero / Maximum downAcceleration
Velocity
b.Determine the following when m = 1 kg, k = 100 N/m and A = 0.25 m.
amaxvmax
T
Kmax
Umax
c.Graph the potential energy, kinetic energy and total energy for one complete oscillation.
3 J0 J
¼T / ¾T
d.complete the following chart (x = 0 at t = 0 s)
t / ¼T / ½T / ¾T / 1Tx
v
a
F
e.Which is the same when the amplitude is increased to 0.50 m?
maximum acceleration / maximum velocity / period73.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?
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?
vava
(A)AC(B)CB
(C)CA(D)DA
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/sd.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.