CT12-1
CTRot- A pocket watch and Big Ben are both keeping perfect time. Which minute hand has the larger angular velocity ω?
A) Pocket watch's
B) Big Ben's
C) Same ω on both.
Which minute hand's tip has the larger tangential velocity?
A) Pocket watch's B) Big Ben's C) Same ω on both.
[pic]Answers: Both Big Ben and the pocket watch have the same magnitude angular velocity [pic]
Big Ben has the larger tangential velocity v = ω r .
[pic]
CTRot- A small wheel and a large wheel are connected by a belt. The small wheel is turned at a constant angular velocity (s. How does the magnitude of the angular velocity of the large wheel (L compare to that of the small wheel?
A) (s = (L B) (s > (L C) (s < (L
There is a bug S on the rim of the small wheel and another bug L on the rim of the large wheel. How do their speeds compare?
A) S = L B) S > L C) S < L
[pic]
Answers: ωs > ωL Every time the big wheel turns once, the little wheel turns several times. So the small wheel turns thru more radians per sec.
S = L. Because the wheels are connected by a belt which does not slip, when one bug moves an inch along the rim, so does the other bug, so they have the same speed.
[pic]
CTRot- A ladybug is clinging to the rim of a spinning wheel which is spinning CCW and is speeding up. At the moment shown, when the bug is at the far right, what is the approximate direction of the ladybug's acceleration?
A)
B) (
C) (
D) (
E)
[pic]
Answer:
The direction of the acceleration a is the same as the direction of Δv. The v1-v2-Δv diagram looks like:
[pic]
[pic]
CTRot- A student in Physics 2010 sees the following question 1 on CAPA due the following Friday night.
An engine flywheel turns with constant angular speed of 100 rev/min. When the engine is shut off, friction slows the wheel to rest in 2 hours. What is the magnitude of the constant angular acceleration of the wheel? Give the answer in units of rev/min2.
The student writes
[pic]
Does the answer come out correctly with the desired units of rev/min2 ?
A) Yes B) No
[pic]Answer: No! You would write the exactly the same equations above if you were trying to get the answer in rad/min2. The factor of 2π in the equation is actually 2π rad/rev, so the units of the answer above are really rad/min2. If you want the answer in rev/min2, you have to leave out the factor of 2π , or you can convert the answer in rad/min2 to rev/min2 by multiplying by the conversion factor (1 rev/2π rad).
[pic]
CTRot- Three forces labeled A, B, C are applied to a rod which pivots on an axis thru its center. [[pic]].
Which force causes the largest magnitude torque?
A B C
D) two or more forces tie for largest size torque.
[pic]
Answer: A.
For C, [pic]. For B, [pic]. For A, [pic].
[pic]
CTRot- A mass is hanging from the end of a horizontal bar which pivots about an axis through it center, but it being held stationary. The bar is released and begins to rotate. As the bar rotates from horizontal to vertical, the magnitude of the torque on the bar..
A) increases B) decreases C) remains constant
As the bar rotates from horizontal to vertical, the magnitude of the angular acceleration ( of the bar... A) increases B) decreases C) remains constant
[pic]
Answers: The torque decreases! [pic]
F⊥ decreases as the mass falls. When the bar is vertical, F⊥ and the torque is zero.
As the bar rotates, the angular acceleration decreases. [pic]. Here I, the moment of inertia, is constant. The torque τ decreases (problem above), so the angular acceleration α must decrease.
Although the angular velocity ω is increasing in magnitude, the angular acceleration is decreasing in magnitude.
[pic]
CTRot- A light rod of length 2L has 2 small heavy masses (each with mass m) attached at the end and the middle. The axis of rotation is at one end, as shown. What is the moment of inertia about the axis?
[pic]
A) 2 m L2 B) 4 m L2 C) 5 m L2 D) 8 m L2
E) None of these
What is the magnitude of the net torque due to gravity?
A) 2mgL B) 3mgL C) 4mgL D) 5mgL
If the bar is released from rest what is the magnitude of the initial angular acceleration?
A) 3g/(5L) B) 5g/(3L) C) 7L/(3g) D) 3L/(5g)
[pic]
Answers:
I = 5 m L2 , [pic]
|τnet | = 3mgL , [pic] (both torques cause rotation in the same sense (CW) so the magnitudes of the torques add.
α = 3g/(5L) [pic]
[pic]
CTRot- A mallet consists of a heavy steel cubical head on a light wooden handle. About which axis of rotation is the moment of inertia I a maximum?
[pic]
[pic]
Answer: Axis B. This axis is farthest from the massive hammer head, so for this axis the moment of inertia [pic] will have the largest ri's .
[pic]
CTRot- Two wheels, each with the same radius R and the same total mass M, are rotating about their fixed axels. Wheel A is a hoop with all the mass very near the rim. Wheel B is a disk with the mass spread out uniformly. Which has the larger moment-of-inertia I = (mi ri2 ?
A) Hoop B) Disk C) both have the same I
[pic]
[pic]
Answer: The hoop has larger I. The hoop has all its mass near the rim, while the disk has its mass spread out between the axis and the rim. [pic], so larger r's for the same total mass results in a larger I.
[pic]
CTRot- A mass m hangs from string wrapped around a pulley of radius R. The pulley has a moment of inertia I and its pivot is frictionless. Because of gravity, the mass accelerates downward and the pulley rotates.
Which has greater magnitude, the string tension FT or the weight mg?
A) FT = mg
B) FT < mg
C) FT > mg
The magnitude of the torque on the pulley is..
A) greater than mgR
B) less than mgR
C) equal to mgR
[pic]
Answers: The string tension is less than mg, and the magnitude of the torque is less than mgR. The tension in the string is less than mg because the mass is accelerating down.
[pic]
Considering the motion of the mass m, we get
[pic]
The tension in the string FT is less than mg. The torque is RFT which is less than Rmg.
[pic]
CTRot- A sphere, a hoop, and a cylinder, all with the same mass M and same radius R, are rolling along, all with the same speed v.
Which has the most total kinetic energy?
A) Sphere B) Hoop C) Disk
D) All have the same KE.
[pic]
Answer: Hoop.
KEtot = KEtrans + KErot = =(1/2)M v2 + (1/2) I ω2.
All three have the same KEtrans=(1/2)M v2. The one with the largest KErot will have the greatest KEtot. That is the one with the largest I (all have the same ω =v/R). IHoop=MR2, IDisk=(1/2)MR2, ISphere=(2/5)MR2. Hoop has the largest I.
You can see this intuitively the following way: The hoop has all its mass at the rim, where things move fastest (speed at the top of the rim is 2v). The others have mass near the middle where things move more slowly.
[pic]
CTRot- Two hoops, A and B, both with the same mass M , are each rotating about an axis thru the center. Hoop B has twice the diameter (twice the R) and is rotating at twice the rate (twice the ω) as hoop A. What is the ration KEB / KEA ?
[pic]
A) 2 B) 4 C) 6 D) 8 E) 16
[pic]
Answer: KE = (1/2) I ω2,
[pic]
[pic]
CTRot- A disk and a hoop, both with the same mass M and the same radius R, both start from rest and roll down an incline. Which one has the largest total kinetic energy at the bottom?
A) Disk B) Hoop C) same KEtot for both
[pic]
[pic]
Answer: Both have the same total KE. Both start out with the same total PE = mgh. For both, all of this initial PE is converted into KE. Although they have the same total KE, the disk has a larger translational KE (and a small rotational KE), so it is moving faster at the bottom.
[pic]
CTRot- Consider a solid disk with an axis of rotation through the center (perpendicular to the diagram). The disk has mass M and radius R. A small mass m is placed on the rim of the disk. What is the moment of inertia of this system?
A) (M+m)R2
B) less than (M+m)R2
C) greater than (M+m)R2
Suppose that mass-disk system is rotating and the axle is frictionless. Atom-Ant carries the mass m toward the center of the rotating disk. As Atom-Ant moves inward, the magnitude of the angular momentum L of the system..
A) increases B) decreases C) remains constant
As Atom-Ant moves inward the kinetic energy of the system..
A) increases B) decreases C) remains constant
(Hint: does Atom-Ant do work? )
Suppose the disk was on a phonograph player, so that it always turned at 33 rpm. As Atom-Ant moves inward, the speed of the mass m A) increases B) decreases C) remains constant
[pic]
Answers:
1) If the system is isolated from outside torques, the total ang.mom. L remains constant. I decreases, ω increases, but L = Iω remains constant.
2) KE increases. Although L = constant, the ant does positive work in pushing the mass toward the center. KErot = (1/2) I ω2 = (1/2)(Iω)(ω). L = Iω remains constant, while ω increases, so KE increases.
3) decreases. Speed v = (2π r)/T. Constant rotation rate means constant period T. As r gets smaller (closer to axis) , v decreases.
Now we can see, qualitatively, why (if the system is isolated) the disk speeds up when the mass m moves toward the center. If the tangential speed v decreases as the mass moves toward the center, there must be a (tangential, not radial) force on the mass causing it to slow down. The disk is exerting a force tending to slow the mass m down. By Newton's third law, there must be a equal and opposite force from the mass m on the disk, tending to speed the disk up.
[pic]
CTRot- Two light (massless) rods, labeled A and B, each are connected to the ceiling by a frictionless pivot. Rod A has length L and has a mass m at the end of the rod. Rod B has length L/2 and has a mass 2m at its end. Both rods are released from rest in a horizontal position.
Which one experiences the larger torque?
A) A B) B C) Both have the same size (.
Which one has the largest moment of inertia I (about the pivot)?
A) A B) B C) Both have the same size I
Which one falls to the vertical position fastest?
A) A B) B C) Both fall at the same rate.
(Hint: [pic])
[pic]
Answers:
Both have the same size torque.
τ A = RFsinθ = Lmg sinθ.
τ B = RFsinθ = (L/2)2mg sinθ= Lmg sinθ
The one with the smallest I has the largest α (since both have the same torque). IA = mL2, IB = 2m(L/2)2 = (1/2) mL2. IB has a smaller I, a larger α , so it falls faster.
[pic]
CTRot- A star is rotating with a period T. Over a period of a million years, its radius decreases by a factor of 2. What is the new period of the star? (Hint) [pic])
A)T/2 B) 2T C) 4T C) T/4 E) None of these.
[pic]
Answer: The new, shorter period is T/4. Angular momentum L = I ω = (2/5)MR2 2π /T = constant, therefore R2/T = constant. If R decreases by a factor of 2, then R2 decreases by a factor of 4, so T must decrease by a factor of 4.
[pic]
CTRot- Consider a solid disk with an axis of rotation through the center (perpendicular to the diagram). The disk has mass M and radius R A small mass m is placed on the rim of the disk. What is the moment of inertia of this system?
A) (M+m)R2
B) less than (M+m)R2
C) greater than (M+m)R2
[pic]
Answer: less than (M+m)R2
Itotal = IDisk + Im = (1/2)MR2 + mR2 = [(1/2)M + m]R2
[pic]
CTRot- Consider two masses, each of size 2m at the ends of a light rod of length L with the axis of rotation through the center of the rod. The rod is doubled in length and the masses are halved. What happens to I?
A) IA > IB B ) IB > IA C) IA = IB
[pic]
Answer: IA < IB
[pic]
[pic]
CTRot- Two wheels with fixed axles, each have the same mass M, but wheel 2 has twice the radius of wheel 1. Each is accelerated from rest with a force applied as shown. Assume that all the mass of the wheels is concentrated in the rims so that the moment of inertia of each is of the form I = M R2 (hoop formula). In order to impart identical angular accelerations to both wheels, how much larger is F2 than F1? Recall that [pic].
A) F2 = F1 . B) F2 = 2F1 . C) F2 = 4F1 . D) F2 = 8F1 .
E) None of these.
[pic]
Answer:
The moment of inertia I of the big hoop is 4 times bigger than the small hoop. So for the big hoop, we need a torque 4 times bigger to get the same angular acceleration. Torque τ = r F(. The larger hoop has a lever arm r twice as big as the small hoop, so we need a force twice as big to get 4 times the torque.
[pic]
CTRot- Consider a rod of uniform density with an axis of rotation through its center and an identical rod with the axis of rotation through one end. Which has the larger moment of inertia?
A) IC > IE B ) IC < IE C ) IC = IE
[pic]
Answer: IE > IC If more of the mass is further from the axis, then the moment of inertia is bigger.[pic]
CTRot- A mass m is attached to a long , massless rod. The mass is close to one end of the rod. Is it easier to balance the rod on end with the mass near the top or near the bottom?
Hint: Small ( means sluggish behavior and [pic].
A) easier with mass near top.
B) easier with mass near bottom.
C) No difference.
[pic]
Answer: the rod with the mass near the top is much easier to balance.
[pic]
So, as r increases, the angular acceleration decreases.
Also, think physically. Big buildings take longer to fall over than do little pencils.
[pic]
-----------------------
[pic]
[pic]
[pic]
[pic]
B
[pic]
M, R
[pic]
[pic]
A
2ω
[pic]
[pic]
[pic]
axis
A
B
C
D
R
M
m
ω
2R
R
R
M
m
m
m
L
L
a
m
R
M, R
hoop
disk
v1
v2
Δv
v1
v2
................
................
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