Question 1

If two forces of equal magnitude act on an object that is hinged at a pivot,the force acting farther from the pivot must produce the greater torque about the pivot.
A) True
B) False
C) unable to decide without knowing the shape of the object

Accepted Answer

B

Question 2

What is the magnitude of the cross product of a vector of magnitude 2.00 m pointing east and a vector of magnitude 4.00 m pointing 30.0° west of north?
A) 6.93
B) -6.93
C) 4.00
D) -4.00
E) 8.00

Accepted Answer

The magnitude of the cross product of two vectors is given by |A x B| = |A||B|sin(θ), where |A| and |B| are the magnitudes of the vectors, and θ is the angle between them. Here, the angle between the eastward vector and the vector pointing 30° west of north (which is 60° east of north) is 90° + 30° = 120°. Thus, |A x B| = 2 * 4 * sin(120°) = 8 * (√3/2) = 4√3 ≈ 6.93.

Question 3

If the torque on an object adds up to zero
A) the forces on it also add up to zero.
B) the object is at rest.
C) the object cannot be turning.
D) the object could be accelerating linearly but it could not be turning.
E) the object could be both turning and accelerating linearly.

Accepted Answer

When the torque on an object adds up to zero, it means there is no net rotational acceleration. However, this does not prevent the object from rotating at a constant rate or from accelerating linearly due to forces that are applied through its center of mass. Thus, an object can indeed be both turning and accelerating linearly under a condition of zero net torque.

Question 4

A solid sphere,solid cylinder,and a hollow pipe all have equal masses and radii and are of uniform density.If the three are released simultaneously at the top of an inclined plane and roll without slipping,which one will reach the bottom first?
A) solid sphere
B) hollow pipe
C) solid cylinder
D) They all reach the bottom at the same time.

Accepted Answer

The solid sphere will reach the bottom first because it has the smallest moment of inertia relative to its shape, which means it will accelerate faster down the incline under the same conditions.

Question 5

A ball is released from rest on a no-slip surface,as shown in the figure.After reaching its lowest point,the ball begins to rise again,this time on a frictionless surface as shown in the figure.When the ball reaches its maximum height on the frictionless surface,it is 
A) at a greater height as when it was released.
B) at a lesser height as when it was released.
C) at the same height as when it was released.
D) It is impossible to tell without knowing the mass of the ball.
E) It is impossible to tell without knowing the radius of the ball.

Accepted Answer

The answer of A ball is released from rest on...

Question 6

If two vectors point in opposite directions,their cross product must be zero.

Accepted Answer

The cross product of two vectors is orthogonal (perpendicular) to both vectors, and its magnitude is given by the product of the magnitudes of the two vectors times the sine of the angle between them. If two vectors point in opposite directions, the angle between them is 180 degrees, and the sine of that angle is zero, which means the magnitude of the cross product is zero. Therefore, the statement is true.

Question 7

A heavy boy and a lightweight girl are balanced on a massless seesaw.If they both move forward so that they are one-half their original distance from the pivot point,what will happen to the seesaw? Assume that both people are small enough compared to the length of the seesaw to be thought of as point masses.
A) It is impossible to say without knowing the masses.
B) It is impossible to say without knowing the distances.
C) The side the boy is sitting on will tilt downward.
D) Nothing will happen; the seesaw will still be balanced.
E) The side the girl is sitting on will tilt downward.

Accepted Answer

The seesaw will remain balanced because both individuals move closer to the pivot point by the same proportion, maintaining the balance due to the principle of moments (torque). The product of their masses and distances from the pivot (mass x distance) remains constant for both, preserving equilibrium.

Question 8

If an irregularly shaped object (such as a wrench)is dropped from rest in a classroom and feels no air resistance,it will
A) accelerate but will not spin.
B) accelerate and turn until its center of gravity reaches its lowest point.
C) accelerate and turn about its center of gravity with uniform angular speed.
D) accelerate and turn about its center of gravity with uniform angular acceleration.
E) accelerate and spin until its center of gravity reaches its highest point.

Accepted Answer

Without air resistance, the only force acting on the object is gravity, which acts uniformly across its mass. This causes the object to accelerate downwards without any rotation or change in orientation, as gravity does not induce spinning or turning on its own.

Question 9

A uniform disk,a uniform hoop,and a uniform solid sphere are released at the same time at the top of an inclined ramp.They all roll without slipping.In what order do they reach the bottom of the ramp?
A) disk, hoop, sphere
B) hoop, sphere, disk
C) sphere, disk, hoop
D) sphere, hoop, disk
E) hoop, disk, sphere

Accepted Answer

The acceleration of each object down the ramp can be found using the formula a = g*sin(theta), where g is the acceleration due to gravity and theta is the angle of the ramp.

For the sphere, the moment of inertia is I = (2/5)*m*r^2, where m is the mass and r is the radius. The torque about the center of mass is τ = I*α, where α is the angular acceleration. For rolling without slipping, τ = F*r, where F is the force of friction. Thus, we can use τ = F*r = (2/5)*m*r^2*α to find that α = (5/7)*g*sin(theta)/r.

For the disk, the moment of inertia is I = (1/2)*m*r^2, so τ = F*r = (1/2)*m*r^2*α and α = (2/5)*g*sin(theta)/r.

For the hoop, the moment of inertia is I = m*r^2, so τ = F*r = m*r^2*α and α = (2/3)*g*sin(theta)/r.

Using the formula v_f^2 = v_i^2 + 2*a*d, where v_f is the final velocity, v_i is the initial velocity (zero), a is the acceleration, and d is the distance down the ramp, we can find that the sphere will have the highest final velocity, followed by the disk and then the hoop. Therefore, the correct order is C) sphere, disk, hoop.

Question 10

If  and 
Are nonzero vectors for which 
• 
= 0,it must follow that
A)  ×
 = 0.
B)  is parallel to
 .
C) |  ×
 | = AB.
D) |  ×
 | = 1.

Accepted Answer

The answer of If  and 
Are nonzero vectors for...

Question 11

If the magnitude of the cross product of two vectors is one-half the dot product of the same vectors,what is the angle between the two vectors?

Accepted Answer

The answer of If the magnitude of the cross product...

Question 12

What is the vector product of  = 2.00 
+ 3)00 
+ 1)00 
And 
= 1)00 
- 3)00 
- 2)00 
?
A) -3.00  + 5.00
 - 9.00

B) -5.00  + 2.00
 - 6.00

C) -9.00  - 3.00
 - 3.00

D) -4.00  + 3.00
 - 1.00

E) 2.00  -9.00
 - 2.00

Accepted Answer

The vector product is found by taking the cross product of the first set of vectors (2i+3j+k) and the second set of vectors (-i-3j-2k). This results in (-2i-5j-9k). Therefore, the coefficients in front of i, j, and k correspond to the components of the resulting vector, giving us -2, -5, -9. So the final answer is -3.00 11ea8401_fb8a_4d4a_aeff_e9ca95f44de4_TB6682_11 + 5.00 11ea8401_fb8a_745b_aeff_2503637e462e_TB6682_11 - 9.00 11ea8401_fb8a_745c_aeff_a5c0da23501a_TB6682_11.

Question 13

When you ride a bicycle,in what direction is the angular velocity of the wheels?
A) to your left
B) to your right
C) forwards
D) backwards
E) up

Accepted Answer

The answer of When you ride a bicycle,in what direction...

Question 14

The angular momentum of a system remains constant
A) when the total kinetic energy is constant.
B) when no net external force acts on the system.
C) when the linear momentum and the energy are constant.
D) when no torque acts on the system.
E) all the time since it is a conserved quantity.

Accepted Answer

The conservation of angular momentum states that the total angular momentum of a system remains constant when there is no external torque acting on it. Therefore, option D, which says that the angular momentum remains constant when no torque acts on the system, is the correct choice. Option E is also technically correct, but less specific. Options A, B, and C are incorrect as they do not address the conservation law for angular momentum.

Question 15

Consider a uniform solid sphere of radius R and mass M rolling without slipping.Which form of its kinetic energy is larger,translational or rotational?
A) Its translational kinetic energy is larger than its rotational kinetic energy.
B) Its rotational kinetic energy is larger than its translational kinetic energy.
C) Both forms of energy are equal.
D) You need to know the speed of the sphere to tell.

Accepted Answer

For a rolling sphere, the total kinetic energy consists of both translational and rotational kinetic energies. 
The translational kinetic energy is given by KE_trans = 1/2 Mv^2, where v is the velocity of the center of mass. 
The rotational kinetic energy is given by KE_rot = 1/2 I ω^2, where I is the moment of inertia of the sphere (I = 2/5 MR^2 for a solid sphere) and ω is the angular velocity of the sphere. 
Since the sphere rolls without slipping, v = Rω, and substituting this in the above equations, we get KE_trans = 1/2 Mv^2 = 1/2 (2/5 MR^2 ω^2) and KE_rot = 1/2 I ω^2 = 1/2 (2/5 MR^2 ω^2). 
Thus, we see that both forms of energy are equal, i.e., KE_trans = KE_rot.

Question 16

A metal bar is hanging from a hook in the ceiling when it is suddenly struck by a ball that is moving horizontally (see figure).The ball is covered with glue,so it sticks to the bar.During this collision 
A) the angular momentum of the system (ball and bar) is conserved about the hook because only gravity is acting on the system.
B) the angular momentum of the system (ball and bar) is not conserved because the hook exerts a force on the bar.
C) the angular momentum of the system (ball and bar) is conserved about the hook because neither the hook nor gravity exerts any torque on this system about the hook.
D) both the angular momentum of the system (ball and bar) and its kinetic energy are conserved.
E) both the linear momentum and the angular momentum of the system (ball and bar) are conserved.

Accepted Answer

The angular momentum of the system (ball and bar) is conserved about the hook because there are no external torques acting on the system about the hook. The only forces acting on the system are gravity and normal forces, which pass through the hook and do not create any torque about it. Therefore, the angular momentum of the system is conserved. Choice A is incorrect because the ball exerts a force on the bar during the collision and there is a torque acting on the system due to this force. Choice B is incorrect because the hook does not exert any torque on the system. Choice D is incorrect because the collision between the ball and the bar is not perfectly elastic, so kinetic energy is not conserved. Choice E is incorrect because the linear momentum of the system is not conserved during the collision because of the sticking together of the ball and the bar.

Question 17

If two vectors are perpendicular to each other,their cross product must be zero.

Accepted Answer

The cross product of two perpendicular vectors is not zero; instead, it results in a vector that is perpendicular to both of the original vectors, with a magnitude equal to the area of the parallelogram that the vectors span.

Question 18

The two rotating systems shown in the figure differ only in that the two identical movable masses are positioned at different distances from the axis of rotation.If you release the hanging blocks simultaneously from rest,and if the ropes do not slip,which block lands first? 
A) The block at the left lands first.
B) The block at the right lands first.
C) Both blocks land at the same time.

Accepted Answer

The block at the right lands first. This is because it has a larger radius of rotation, which means it has a greater linear velocity and takes longer to complete one full rotation. Therefore, it will travel a greater distance and land first.

Question 19

A tire is rolling along a road,without slipping,with a velocity v.A piece of tape is attached to the tire.When the tape is opposite the road (at the top of the tire),its velocity with respect to the road is
A) 2v.
B) v.
C) 1.5v.
D) zero.
E) The velocity depends on the radius of the tire.

Accepted Answer

The velocity of the tape when it is at the top of the tire is 2v because the tire is rolling without slipping. The center of the tire moves with velocity v, and since the tire is rotating, the top point moves with an additional velocity v relative to the center, resulting in a total velocity of 2v relative to the road.

Question 20

As you are leaving a building,the door opens outward.If the hinges on the door are on your right,what is the direction of the angular velocity of the door as you open it?
A) up
B) down
C) to your left
D) to your right
E) forwards

Accepted Answer

The answer of As you are leaving a building,the door...

Quiz 12: Torque

If two forces of equal magnitude act on an object that is hinged at a pivot,the force acting farther from the pivot must produce the greater torque about the pivot.

What is the magnitude of the cross product of a vector of magnitude 2.00 m pointing east and a vector of magnitude 4.00 m pointing 30.0° west of north?

If the torque on an object adds up to zero

A solid sphere,solid cylinder,and a hollow pipe all have equal masses and radii and are of uniform density.If the three are released simultaneously at the top of an inclined plane and roll without slipping,which one will reach the bottom first?

A ball is released from rest on a no-slip surface,as shown in the figure.After reaching its lowest point,the ball begins to rise again,this time on a frictionless surface as shown in the figure.When the ball reaches its maximum height on the frictionless surface,it is

If two vectors point in opposite directions,their cross product must be zero.

If an irregularly shaped object (such as a wrench) is dropped from rest in a classroom and feels no air resistance,it will

A uniform disk,a uniform hoop,and a uniform solid sphere are released at the same time at the top of an inclined ramp.They all roll without slipping.In what order do they reach the bottom of the ramp?

If and Are nonzero vectors for which • = 0,it must follow that

If the magnitude of the cross product of two vectors is one-half the dot product of the same vectors,what is the angle between the two vectors?

What is the vector product of = 2.00 + 3) 00 + 1) 00 And = 1) 00 - 3) 00 - 2) 00 ?

When you ride a bicycle,in what direction is the angular velocity of the wheels?

The angular momentum of a system remains constant

Consider a uniform solid sphere of radius R and mass M rolling without slipping.Which form of its kinetic energy is larger,translational or rotational?

A metal bar is hanging from a hook in the ceiling when it is suddenly struck by a ball that is moving horizontally (see figure) .The ball is covered with glue,so it sticks to the bar.During this collision

If two vectors are perpendicular to each other,their cross product must be zero.

The two rotating systems shown in the figure differ only in that the two identical movable masses are positioned at different distances from the axis of rotation.If you release the hanging blocks simultaneously from rest,and if the ropes do not slip,which block lands first?

A tire is rolling along a road,without slipping,with a velocity v.A piece of tape is attached to the tire.When the tape is opposite the road (at the top of the tire) ,its velocity with respect to the road is

As you are leaving a building,the door opens outward.If the hinges on the door are on your right,what is the direction of the angular velocity of the door as you open it?