Question 1

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  the other rotates in the opposite direction at 3.42 rad/s.Calculate the magnitude of the net angular momentum of the system.

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Question 2

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?

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 3

A 3.00-kg ball rests in a frictionless groove as shown in the figure.  (a)What is the magnitude of the force that the left side of the groove exerts on the ball?
(b)What is the magnitude of the force that the right side of the groove exerts on the ball?

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Question 4

A bicycle wheel of radius 0.36 m and mass 3.2 kg is set spinning at 4.00 rev/s.A very light bolt is attached to extend the axle in length,and a string is attached to the axle at a distance of 0.10 m from the wheel.Initially the axle of the spinning wheel is horizontal,and the wheel is suspended only from the string.We can ignore the mass of the axle and spokes.At what rate will the wheel process about the vertical?

Accepted Answer

The correct answer is A.Explanation:The initial angular momentum of the wheel is:$$L_i = I_i \omega_i = \frac{1}{2}mr^2 \omega_i$$where:* $I_i$ is the initial moment of inertia of the wheel* $m$ is the mass of the wheel* $r$ is the radius of the wheel* $\omega_i$ is the initial angular velocity of the wheelWhen the bolt is attached, the moment of inertia of the wheel increases to:$$I_f = I_i + mr_b^2$$where:* $r_b$ is the distance from the axle to the boltThe final angular velocity of the wheel is:$$\omega_f = \frac{L_i}{I_f} = \frac{\frac{1}{2}mr^2 \omega_i}{\frac{1}{2}mr^2 + mr_b^2}$$Substituting the given values, we get:$$\omega_f = \frac{\frac{1}{2}(3.2 	ext{ kg})(0.36 	ext{ m})^2 (4.00 	ext{ rev/s})}{\frac{1}{2}(3.2 	ext{ kg})(0.36 	ext{ m})^2 + (3.2 	ext{ kg})(0.10 	ext{ m})^2} = 2.9 	ext{ rev/s}$$Therefore, the wheel will process about the vertical at a rate of 2.9 rpm.

Question 5

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?

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Question 6

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?

Accepted Answer

Conservation of angular momentum states that the angular momentum of a system remains constant if no external torque acts on it. Therefore, initial angular momentum (Iω) is equal to final angular momentum (I'ω').

Iω = I'ω'

We can rearrange this equation to solve for the final angular velocity:

ω' = (Iω)/I'

Substituting the given values, we get:

ω' = (2.25 kg ∙ m2/s)(5.00 rad/s)/(1.80 kg ∙ m2/s)
ω' = 6.25 rad/s

Therefore, the final angular speed is 6.25 rad/s, which is choice C.

Question 7

A 30.0-kg child sits on one end of a long uniform beam having a mass of 20.0 kg,and a 40.0-kg child sits on the other end.The beam balances when a fulcrum is placed below the beam a distance 1.10 m from the 30.0-kg child.How long is the beam?

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Question 8

A uniform solid disk of radius 1.60 m and mass 2.30 kg rolls without slipping to the bottom of an inclined plane.If the angular velocity of the disk is  at the bottom,what is the height of the inclined plane?

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Question 9

An object is rotating with an angular momentum 4.00 kg ∙ m²/s while being acted on by a constant torque 3.00 N ∙ m .What is the angular speed of precession of the angular velocity of the object?

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Question 10

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?

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Question 11

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?

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 of the wheel moves towards the north, the angular momentum vector points towards the west.

Question 12

An 82.0 kg-diver stands at the edge of a light 5.00-m diving board,which is supported by two narrow pillars 1.60 m apart,as shown in the figure.Find the magnitude and direction of the force exerted on the diving board
(a)by pillar A.
(b)by pillar B.

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Question 13

A nonuniform,80.0-g,meterstick balances when the support is placed at the 51.0-cm mark.At what location on the meterstick should a 5.00-g tack be placed so that the stick will balance at the 50.0 cm mark?

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Question 14

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?

Accepted Answer

Using the conservation of angular momentum, the initial angular momentum (I_initial * ω_initial) equals the final angular momentum ((I_initial + m * r^2) * ω_final). Solving for m gives 8.0 kg.

Question 15

In the figure,the rotor of a gyroscope is a uniform disk that has a radius of 13.8 cm and a moment of inertia about its axis of 4.0 × 10^-3 kg ∙ m².The length of the rotor shaft is 6.9 cm.The ball pivot at P is frictionless.At a given instant,the rotor shaft is horizontal and the rotor is rotating with an angular velocity of 90 rad/s about its axis OP,as shown.The rotor is viewed from point Q on the axis.The precessional linear velocity of point O,including the direction seen from point Q,is closest to

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Question 16

A 500-g particle is located at the point  = 4m  + 3m  - 2m  and is moving with a velocity  = 5 m/s  -2 ms  + 4 m/s  .What is the angular momentum of this particle about the origin?

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Question 17

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?

Accepted Answer

The conservation of angular momentum can be applied to the system of the turntable and the ball. Before the ball is caught, the angular momentum of the system is:
L = Iω = (2.0 kg · m²)(1.5 rad/s) = 3.0 kg · m²/s
After the ball is caught, the turntable will experience a slight increase in its moment of inertia due to the added mass of the ball. However, the effect of this increase is negligible compared to the effect of the ball's angular momentum. The ball's angular momentum is:
L_ball = mvr = (0.40 kg)(0.80 m)(3.0 m/s) = 0.96 kg · m²/s
Since the turntable is initially rotating about a vertical axis through its center, the angular momentum of the ball is initially in the horizontal direction. When the ball is caught by the mechanism on the rim of the turntable, its angular momentum is transfered to the turntable in the vertical direction. Therefore, the final angular momentum of the system is:
L_f = L + L_ball = 3.96 kg · m²/s
The final angular velocity of the turntable can be calculated using the final angular momentum and the moment of inertia of the system:
L_f = I_fω_f
ω_f = L_f / I_f
where I_f = I + mR² is the moment of inertia of the system after the ball is caught, where R is the radius of the turntable. Substituting the given values, we get:
ω_f = (3.96 kg · m²/s) / (2.0 kg · m² + 0.40 kg * (0.80 m)²) = 1.27 rad/s
Therefore, the angular velocity of the turntable just after the ball is caught is 1.3 rad/s (to two significant figures), which corresponds to choice (B).

Question 18

A 20.0-kg uniform plank is supported by the floor at one end and by a vertical rope at the other as shown in the figure.A 50.0-kg mass person stands on the plank a distance three-fourths of the length plank from the end on the floor.  (a)What is the tension in the rope?
(b)What is the magnitude of the force that the floor exerts on the plank?

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Question 19

A light board,10 m long,is supported by two sawhorses,one at one edge of the board and a second at the midpoint.A small 40-N weight is placed between the two sawhorses,3.0 m from the edge and 2.0 m from the center.What forces are exerted by the sawhorses on the board?

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Question 20

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

Accepted Answer

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

Quiz 12: Rotation of a Rigid Body

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 the other rotates in the opposite direction at 3.42 rad/s.Calculate the magnitude of the net angular momentum of the system.

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 3.00-kg ball rests in a frictionless groove as shown in the figure. (a)What is the magnitude of the force that the left side of the groove exerts on the ball? (b)What is the magnitude of the force that the right side of the groove exerts on the ball?

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 30.0-kg child sits on one end of a long uniform beam having a mass of 20.0 kg,and a 40.0-kg child sits on the other end.The beam balances when a fulcrum is placed below the beam a distance 1.10 m from the 30.0-kg child.How long is the beam?

A uniform solid disk of radius 1.60 m and mass 2.30 kg rolls without slipping to the bottom of an inclined plane.If the angular velocity of the disk is at the bottom,what is the height of the inclined plane?

An object is rotating with an angular momentum 4.00 kg ∙ m²/s while being acted on by a constant torque 3.00 N ∙ m .What is the angular speed of precession of the angular velocity of the object?

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?

An 82.0 kg-diver stands at the edge of a light 5.00-m diving board,which is supported by two narrow pillars 1.60 m apart,as shown in the figure.Find the magnitude and direction of the force exerted on the diving board (a)by pillar A. (b)by pillar B.

A nonuniform,80.0-g,meterstick balances when the support is placed at the 51.0-cm mark.At what location on the meterstick should a 5.00-g tack be placed so that the stick will balance at the 50.0 cm mark?

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 500-g particle is located at the point = 4m + 3m - 2m and is moving with a velocity = 5 m/s -2 ms + 4 m/s .What is the angular momentum of this particle about the origin?

A light board,10 m long,is supported by two sawhorses,one at one edge of the board and a second at the midpoint.A small 40-N weight is placed between the two sawhorses,3.0 m from the edge and 2.0 m from the center.What forces are exerted by the sawhorses on the board?

Quiz 12: Rotation of a Rigid Body

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 the other rotates in the opposite direction at 3.42 rad/s.Calculate the magnitude of the net angular momentum of the system.

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 3.00-kg ball rests in a frictionless groove as shown in the figure. (a)What is the magnitude of the force that the left side of the groove exerts on the ball? (b)What is the magnitude of the force that the right side of the groove exerts on the ball?

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 30.0-kg child sits on one end of a long uniform beam having a mass of 20.0 kg,and a 40.0-kg child sits on the other end.The beam balances when a fulcrum is placed below the beam a distance 1.10 m from the 30.0-kg child.How long is the beam?

A uniform solid disk of radius 1.60 m and mass 2.30 kg rolls without slipping to the bottom of an inclined plane.If the angular velocity of the disk is at the bottom,what is the height of the inclined plane?

An object is rotating with an angular momentum 4.00 kg ∙ m2/s while being acted on by a constant torque 3.00 N ∙ m .What is the angular speed of precession of the angular velocity of the object?

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?

An 82.0 kg-diver stands at the edge of a light 5.00-m diving board,which is supported by two narrow pillars 1.60 m apart,as shown in the figure.Find the magnitude and direction of the force exerted on the diving board (a)by pillar A. (b)by pillar B.

A nonuniform,80.0-g,meterstick balances when the support is placed at the 51.0-cm mark.At what location on the meterstick should a 5.00-g tack be placed so that the stick will balance at the 50.0 cm mark?

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 500-g particle is located at the point = 4m + 3m - 2m and is moving with a velocity = 5 m/s -2 ms + 4 m/s .What is the angular momentum of this particle about the origin?

A light board,10 m long,is supported by two sawhorses,one at one edge of the board and a second at the midpoint.A small 40-N weight is placed between the two sawhorses,3.0 m from the edge and 2.0 m from the center.What forces are exerted by the sawhorses on the board?

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?

An object is rotating with an angular momentum 4.00 kg ∙ m²/s while being acted on by a constant torque 3.00 N ∙ m .What is the angular speed of precession of the angular velocity of the object?

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?