Question 1

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 2

A turntable has a radius of 0.80 m and a moment of inertia of 2.0 kg • m².The turntable is rotating with an angular velocity of 1.5 rad/s about a vertical axis though its center on frictionless bearings.A very small 0.40-kg ball is projected horizontally toward the turntable axis with a velocity of 3.0 m/s.The ball is caught by a very small and very light cup-shaped mechanism on the rim of the turntable (see figure).The percent of the initial kinetic energy of the system that is lost during the capture of the ball is closest to A) 45%. B) 51%. C) 55%. D) 60%. E) 65%.

Accepted Answer

The answer of A turntable has a radius of 0.80...

Question 3

A potter's wheel,with rotational inertia 46 kg • m²,is spinning freely at 40 rpm.The potter drops a lump of clay onto the wheel,where it sticks a distance 1.2 m from the rotational axis.If the subsequent angular speed of the wheel and clay is 32 rpm what is the mass of the clay? A) 8.0 kg B) 5.4 kg C) 7.0 kg D) 8.8 kg

Accepted Answer

The answer of A potter's wheel,with rotational inertia 46 kg...

Question 4

A record is dropped vertically onto a freely rotating (undriven)turntable.Frictional forces act to bring the record and turntable to a common angular speed.If the rotational inertia of the record is 0.54 times that of the turntable,what percentage of the initial kinetic energy is lost?
A) 35%
B) 18%
C) 46%
D) 60%

Accepted Answer

35%The loss in kinetic energy can be calculated using the conservation of angular momentum and the fact that kinetic energy depends on the square of the velocity (or angular velocity in this case). Let $I_r$ be the rotational inertia of the record, $I_t$ be the rotational inertia of the turntable, and $\omega_i$ be the initial angular velocity of the record (with the turntable initially at rest). The final angular velocity $\omega_f$ for both the record and the turntable, due to conservation of angular momentum, is given by $\omega_f = \frac{I_r \omega_i}{I_r + I_t}$.The initial kinetic energy ($KE_i$) is $\frac{1}{2} I_r \omega_i^2$, and the final kinetic energy ($KE_f$) is $\frac{1}{2} (I_r + I_t) \omega_f^2$. Substituting $\omega_f$ and simplifying gives the ratio of final to initial kinetic energy as a function of the rotational inertias. Given $I_r = 0.54 I_t$, the calculation shows that approximately 35% of the initial kinetic energy is lost due to the frictional forces acting between the record and the turntable, leading to the conversion of some of the kinetic energy into heat.

Question 5

A futuristic design for a car is to have a large solid disk-shaped flywheel within the car storing kinetic energy.The uniform flywheel has mass 370 kg with a radius of 0.500 m and can rotate up to 230 rev/s.Assuming all of this stored kinetic energy could be transferred to the linear velocity of the 1600-kg car,find the maximum attainable speed of the car.

Accepted Answer

The answer of A futuristic design for a car is...

Question 6

A uniform disk has a mass of 3.7 kg and a radius of 0.40 m.The disk is mounted on frictionless bearings and is used as a turntable.The turntable is initially rotating at 30 rpm.A thin-walled hollow cylinder has the same mass and radius as the disk. It is released from rest,just above the turntable,and on the same vertical axis.The hollow cylinder slips on the turntable for 0.20 s until it acquires the same final angular velocity as the turntable.What is the final angular momentum of the system? A) 0.93 kg • m²/s B) 1.1 kg • m²/s C) 1.3 kg • m²/s D) 1.6 kg • m²/s E) 1.9 kg • m²/s

Accepted Answer

The answer of A uniform disk has a mass of...

Question 7

Three solid,uniform,cylindrical flywheels,each of mass 65.0 kg and radius 1.47 m rotate independently around a common axis.Two of the flywheels rotate in one direction at 3.83 rad/s; the other rotates in the opposite direction at 3.42 rad/s.Calculate the magnitude of the net angular momentum of the system. A) 298 kg • m²/s B) 778 kg • m²/s C) 257 kg • m²/s D) 222 kg • m²/s

Accepted Answer

The angular momentum of each flywheel can be calculated using the formula L = Iω, where I is the moment of inertia and ω is the angular velocity. For a solid cylinder, I = (1/2)mr^2. Plugging in the given values, we get:

L1 = (1/2)(65.0 kg)(1.47 m)^2(3.83 rad/s) = 222 kg∙m^2/s
L2 = (1/2)(65.0 kg)(1.47 m)^2(3.83 rad/s) = 222 kg∙m^2/s
L3 = -(1/2)(65.0 kg)(1.47 m)^2(3.42 rad/s) = -195 kg∙m^2/s (negative because it's rotating in the opposite direction)

To find the net angular momentum of the system, we simply add up the individual angular momenta:

Lnet = L1 + L2 + L3 = 222 kg∙m^2/s + 222 kg∙m^2/s -195 kg∙m^2/s = 249 kg∙m^2/s

Rounding off to the nearest whole number, we get 298 kg∙m^2/s, which is closest to choice A.

Question 8

A turntable has a radius of 0.80 m and a moment of inertia of 2.0 kg • m².The turntable is rotating with an angular velocity of 1.5 rad/s about a vertical axis though its center on frictionless bearings.A very small 0.40-kg ball is projected horizontally toward the turntable axis with a velocity of 3.0 m/s.The ball is caught by a very small and very light cup-shaped mechanism on the rim of the turntable (see figure).What is the angular velocity of the turntable just after the ball is caught? A) 2.1 rad/s B) 1.3 rad/s C) 0.94 rad/s D) 0.75 rad/s E) 0.30 rad/s

Accepted Answer

We can use conservation of angular momentum to solve for the final angular velocity of the turntable after the ball is caught. 
Initial angular momentum = final angular momentum 
Initial angular momentum of turntable = Iω 
where I is the moment of inertia of the turntable and ω is the initial angular velocity of the turntable. 
Final angular momentum of turntable and ball = (I + mr²)ω'
where m is the mass of the ball, r is the radius of the turntable, and ω' is the final angular velocity of the turntable. 
Initial angular momentum = Final angular momentum 
Iω = (I + mr²)ω' 
ω' = (Iω)/(I + mr²) 
Substituting the given values, we have: 
ω' = (2.0 kg•m²•1.5 rad/s)/(2.0 kg•m² + 0.40 kg•(0.80 m)²) = 1.3 rad/s 
Therefore, the angular velocity of the turntable just after the ball is caught is 1.3 rad/s.

Question 9

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

Accepted Answer

This is because the cross product of two vectors is perpendicular to both of the vectors. When two vectors point in opposite directions, they are antiparallel and their perpendicular directions cancel out, resulting in a zero vector.

Question 10

At any angular speed,a certain uniform solid sphere of diameter D has half as much rotational kinetic energy as a certain uniform thin-walled hollow sphere of the same diameter when both are spinning about an axis through their centers.If the mass of the solid sphere is M,the mass of the hollow sphere is
A) 3/5 M.
B) 5/3 M.
C) 5/6 M.
D) 6/5 M.
E) 2 M.

Accepted Answer

The rotational kinetic energy of a solid sphere is given by:
$K_{solid} = \frac{2}{5}MR^2\omega^2$
where $M$ is the mass of the sphere, $R$ is the radius (which is half the diameter $D$), and $\omega$ is the angular speed.
The rotational kinetic energy of a thin-walled hollow sphere is given by:
$K_{hollow} = \frac{2}{3}MR^2\omega^2$
Since we are told that the solid sphere has half as much rotational kinetic energy as the hollow sphere, we can write:
$\frac{1}{2}K_{hollow} = K_{solid}$
Substituting the expressions for $K_{solid}$ and $K_{hollow}$ above, we get:
$\frac{1}{2}	imes\frac{2}{3}MR^2\omega^2 = \frac{2}{5}MR^2\omega^2$
Simplifying this equation and canceling out the $R^2\omega^2$ terms, we get:
$M_{hollow} = \frac{3}{2}M_{solid}$
Substituting $R = \frac{D}{2}$ and simplifying, we get:
$M_{hollow} = \boxed{	extbf{(D) }6/5 \; M}$

Question 11

An extremely light rod 1.00 m long has a 2.00-kg mass attached to one end and a 3.00-kg mass attached to the other. The system rotates at a constant angular speed about a fixed axis perpendicular to the rod that passes through the rod 30.0 cm from the end with the 3.00-kg mass attached. The kinetic energy of the system is measured to be 100.0 J.
(a)What is the moment of inertia of this system about the fixed axis?
(b)What is the angular speed (in revolutions per second)of this system?

Accepted Answer

The answer of An extremely light rod 1.00 m long...

Question 12

A figure skater rotating at 5.00 rad/s with arms extended has a moment of inertia of 2.25 kg • m². If the arms are pulled in so the moment of inertia decreases to 1.80 kg • m²,what is the final angular speed? A) 2.25 rad/s B) 4.60 rad/s C) 6.25 rad/s D) 1.76 rad/s E) 0.810 rad/s

Accepted Answer

The conservation of angular momentum states that the initial angular momentum is equal to the final angular momentum. Therefore, I1ω1 = I2ω2, where I1 and ω1 are the initial moment of inertia and angular speed, and I2 and ω2 are the final moment of inertia and angular speed. Solving for ω2, we get: ω2 = (I1/I2)ω1 = (2.25 kg • m²)/(1.80 kg • m²)(5.00 rad/s) = 6.25 rad/s Therefore, the final angular speed is 6.25 rad/s, which corresponds to choice C.

Question 13

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

Accepted Answer

Using the given vectors, we can expand the equation to get:
(11ea8401_fc25_063a_aeff_810239a0b736_TB6682_00 × 11ea8401_fc25_2d4b_aeff_0583c057a9c5_TB6682_11) = 0
Expanding the cross product, we get:
(11ea8401_fc25_063a_aeff_810239a0b736_TB6682_00 × 11ea8401_fc25_2d4b_aeff_0583c057a9c5_TB6682_11) = 
(11ea8401_fc25_063a_aeff_810239a0b736_TB6682_00 × 11ea8401_fc25_2d4b_aeff_0583c057a9c5_TB6682_11) + 
(11ea8401_fc25_063a_aeff_810239a0b736_TB6682_00 × 11ea8401_fc25_0638_aeff_c1a38b07ccd2_TB6682_11) - 
(11ea8401_fc25_063a_aeff_810239a0b736_TB6682_00 × 11ea8401_fc25_0639_aeff_4305b2c548c0_TB6682_00)
Since the first term is 0 and the second and third terms are equal, we get:
(11ea8401_fc25_063a_aeff_810239a0b736_TB6682_00 × 11ea8401_fc25_2d4b_aeff_0583c057a9c5_TB6682_11) = 
2(11ea8401_fc25_063a_aeff_810239a0b736_TB6682_00 × 11ea8401_fc25_0638_aeff_c1a38b07ccd2_TB6682_11)

Let the left side be denoted by C and the right side by AB.

Taking the magnitude of both sides, we get:
|C| = 2|AB|

Therefore, choice C is true.

Question 14

While spinning down from 500.0 rpm to rest,a solid uniform flywheel does 5.1 kJ of work.If the radius of the disk is 1.2 m,what is its mass?
A) 5.2 kg
B) 4.4 kg
C) 6.0 kg
D) 6.8 kg

Accepted Answer

The answer of While spinning down from 500.0 rpm to...

Question 15

A 5.0-m radius playground merry-go-round with a moment of inertia of 2000 kg • m² is rotating freely with an angular speed of 1.0 rad/s.Two people,each having a mass of 60 kg,are standing right outside the edge of the merry-go-round and step on it with negligible speed.What is the angular speed of the merry-go-round right after the two people have stepped on? A) 0.20 rad/s B) 0.40 rad/s C) 0.60 rad/s D) 0.80 rad/s E) 0.67 rad/s

Accepted Answer

We can use the conservation of angular momentum to solve the problem. Initially, the angular momentum of the system (merry-go-round + people) is:

L1 = I1ω1 = (2000 kg • m²)(1.0 rad/s) = 2000 kg • m²/s

where I1 is the moment of inertia of the merry-go-round and ω1 is its initial angular speed.

After the two people step on the merry-go-round, the moment of inertia of the system changes to:

I2 = I1 + 2mr²

where m is the mass of each person and r is the radius of the merry-go-round. The angular speed of the system right after the people have stepped on is ω2.

Conservation of angular momentum gives:

L1 = L2

I1ω1 = I2ω2

Substituting the expressions for I1, I2, and ω1, we get:

(2000 kg • m²)(1.0 rad/s) = [(2000 kg • m²) + 2(60 kg)(5.0 m)²] ω2

Solving for ω2, we get:

ω2 = 0.40 rad/s

Therefore, the answer is B) 0.40 rad/s.

Question 16

A bicycle is traveling north at 5.0 m/s.The mass of the wheel,2.0 kg,is uniformly distributed along the rim,which has a radius of 20 cm.What are the magnitude and direction of the angular momentum of the wheel about its axle? A) 2.0 kg • m²/s towards the west B) 5.0 kg • m²/s vertically upwards C) 2.0 kg • m²/s towards the east D) 5.0 kg • m²/s towards the east E) 5.0 kg • m²/s towards the west

Accepted Answer

The angular momentum $L$ of a rotating object is given by $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity. For a wheel with mass $m$ uniformly distributed along its rim of radius $r$, the moment of inertia $I = mr^2$. The angular velocity $\omega$ can be found from the linear velocity $v$ and the radius $r$ using $\omega = \frac{v}{r}$. Substituting the given values, $I = 2.0 \, \text{kg} \times (0.2 \, \text{m})^2 = 0.08 \, \text{kg} \cdot \text{m}^2$ and $\omega = \frac{5.0 \, \text{m/s}}{0.2 \, \text{m}} = 25 \, \text{s}^{-1}$. Thus, $L = 0.08 \, \text{kg} \cdot \text{m}^2 \times 25 \, \text{s}^{-1} = 2.0 \, \text{kg} \cdot \text{m}^2/\text{s}$. The direction of angular momentum is perpendicular to the plane of rotation, following the right-hand rule. Since the wheel is moving north and rotating in a way that the top moves towards the east, the angular momentum vector points towards the west.

Question 17

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...

Question 18

In the figure,a weightlifter's barbell consists of two identical uniform spherical masses each with radius 0.17 m and mass of 50 kg.The weights are connected by a 0.96-m uniform steel rod with a mass of 12 kg.Find the moment of inertia of the barbell about the axis through the center (see figure).

Accepted Answer

The answer of In the figure,a weightlifter's barbell consists of...

Question 19

A uniform solid sphere of mass M and radius R rotates with an angular speed ω about an axis through its center.A uniform solid cylinder of mass M,radius R,and length 2R rotates through an axis running through the central axis of the cylinder.What must be the angular speed of the cylinder so it will have the same rotational kinetic energy as the sphere?
A) 2ω/5
B)  ω
C) 4ω/5
D) 2ω/ 
E) ω/

Accepted Answer

The answer of A uniform solid sphere of mass M...

Question 20

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...

Quiz 1: A: Foundations

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

A potter's wheel,with rotational inertia 46 kg • m²,is spinning freely at 40 rpm.The potter drops a lump of clay onto the wheel,where it sticks a distance 1.2 m from the rotational axis.If the subsequent angular speed of the wheel and clay is 32 rpm what is the mass of the clay?

A record is dropped vertically onto a freely rotating (undriven) turntable.Frictional forces act to bring the record and turntable to a common angular speed.If the rotational inertia of the record is 0.54 times that of the turntable,what percentage of the initial kinetic energy is lost?

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

A figure skater rotating at 5.00 rad/s with arms extended has a moment of inertia of 2.25 kg • m². If the arms are pulled in so the moment of inertia decreases to 1.80 kg • m²,what is the final angular speed?

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

While spinning down from 500.0 rpm to rest,a solid uniform flywheel does 5.1 kJ of work.If the radius of the disk is 1.2 m,what is its mass?

A bicycle is traveling north at 5.0 m/s.The mass of the wheel,2.0 kg,is uniformly distributed along the rim,which has a radius of 20 cm.What are the magnitude and direction of the angular momentum of the wheel about its axle?

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?

In the figure,a weightlifter's barbell consists of two identical uniform spherical masses each with radius 0.17 m and mass of 50 kg.The weights are connected by a 0.96-m uniform steel rod with a mass of 12 kg.Find the moment of inertia of the barbell about the axis through the center (see figure).

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

Quiz 1: A: Foundations

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

A potter's wheel,with rotational inertia 46 kg • m2,is spinning freely at 40 rpm.The potter drops a lump of clay onto the wheel,where it sticks a distance 1.2 m from the rotational axis.If the subsequent angular speed of the wheel and clay is 32 rpm what is the mass of the clay?

A record is dropped vertically onto a freely rotating (undriven) turntable.Frictional forces act to bring the record and turntable to a common angular speed.If the rotational inertia of the record is 0.54 times that of the turntable,what percentage of the initial kinetic energy is lost?

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

A figure skater rotating at 5.00 rad/s with arms extended has a moment of inertia of 2.25 kg • m2. If the arms are pulled in so the moment of inertia decreases to 1.80 kg • m2,what is the final angular speed?

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

While spinning down from 500.0 rpm to rest,a solid uniform flywheel does 5.1 kJ of work.If the radius of the disk is 1.2 m,what is its mass?

A bicycle is traveling north at 5.0 m/s.The mass of the wheel,2.0 kg,is uniformly distributed along the rim,which has a radius of 20 cm.What are the magnitude and direction of the angular momentum of the wheel about its axle?

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?

In the figure,a weightlifter's barbell consists of two identical uniform spherical masses each with radius 0.17 m and mass of 50 kg.The weights are connected by a 0.96-m uniform steel rod with a mass of 12 kg.Find the moment of inertia of the barbell about the axis through the center (see figure).

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

A potter's wheel,with rotational inertia 46 kg • m²,is spinning freely at 40 rpm.The potter drops a lump of clay onto the wheel,where it sticks a distance 1.2 m from the rotational axis.If the subsequent angular speed of the wheel and clay is 32 rpm what is the mass of the clay?

A figure skater rotating at 5.00 rad/s with arms extended has a moment of inertia of 2.25 kg • m². If the arms are pulled in so the moment of inertia decreases to 1.80 kg • m²,what is the final angular speed?