A.4 Rigid body mechanics

Two objects of mass $M$ each are connected by a weightless rod of length $d$. A force $F$ is applied to each of the objects, at right angles to the rod as shown.

What is the torque acting on the system about the midpoint of the rod and what is the angular acceleration of the system?

Two objects of mass $M$ each are connected by a weightless rod of length $d$. A force $F$ is applied to each of the objects, at right angles to the rod as shown.

What is the torque acting on the system about the midpoint of the rod and what is the angular acceleration of the system?

A turntable of mass $M$ and radius $R$ spins freely about the vertical axis at an initial angular velocity $\omega$. The moment of inertia of the turntable about the axis of rotation is $\frac{1}{2}M{R}^2$. A small body of mass $m$ is dropped close to the edge of the turntable with a negligible initial velocity.

The body comes to rest relative to the turntable. What is the final angular velocity of the turntable?

A.  $\frac{M}{M+2m}\omega$

B.  $\frac{M}{M+m}\omega$

C.  $\frac{M}{2M+m}\omega$

D.  $\frac{2M}{M+m}\omega$

A turntable of mass $M$ and radius $R$ spins freely about the vertical axis at an initial angular velocity $\omega$. The moment of inertia of the turntable about the axis of rotation is $\frac{1}{2}M{R}^2$. A small body of mass $m$ is dropped close to the edge of the turntable with a negligible initial velocity.

The body comes to rest relative to the turntable. What is the final angular velocity of the turntable?

A.  $\frac{M}{M+2m}\omega$

B.  $\frac{M}{M+m}\omega$

C.  $\frac{M}{2M+m}\omega$

D.  $\frac{2M}{M+m}\omega$

Show that the rotational kinetic energy of the turbine decreases at a constant rate.

Show that the rotational kinetic energy of the turbine decreases at a constant rate.

Show that the rotational kinetic energy of the turbine decreases at a constant rate.

Calculate the angular impulse delivered to the flywheel during the acceleration.

Calculate the angular impulse delivered to the flywheel during the acceleration.

Calculate the angular impulse delivered to the flywheel during the acceleration.

Determine the average magnitude of $F$.

Determine the average magnitude of $F$.

Determine the average magnitude of $F$.

State two assumptions of your calculation in part (b).

State two assumptions of your calculation in part (b).

State two assumptions of your calculation in part (b).

the angular acceleration of the ring;

the angular acceleration of the ring;

the angular acceleration of the ring;

the angular velocity of the ring after a time of 5.0 s.

the angular velocity of the ring after a time of 5.0 s.

the angular velocity of the ring after a time of 5.0 s.

the final kinetic energy of the disc and the ring.

the final kinetic energy of the disc and the ring.

the final kinetic energy of the disc and the ring.

For the propellor, $L=5.0\text{\hspace{0.05em}cm}$ and $M=0.035 \text{kg}$.

Calculate the moment of inertia of the propellor.

For the propellor, $L=5.0\text{\hspace{0.05em}cm}$ and $M=0.035 \text{kg}$.

Calculate the moment of inertia of the propellor.

For the propellor, $L=5.0\text{\hspace{0.05em}cm}$ and $M=0.035 \text{kg}$.

Calculate the moment of inertia of the propellor.

Calculate the angular impulse that acts on the propellor.

Calculate the angular impulse that acts on the propellor.

Calculate the angular impulse that acts on the propellor.

State and explain the effect of the angular impulse on the body of the aeroplane.

State and explain the effect of the angular impulse on the body of the aeroplane.

State and explain the effect of the angular impulse on the body of the aeroplane.

The angular speed of the flywheel increased by 280 rad s⁻¹ during the application of the angular impulse.

Determine the moment of inertia of the flywheel.

The angular speed of the flywheel increased by 280 rad s⁻¹ during the application of the angular impulse.

Determine the moment of inertia of the flywheel.

The angular speed of the flywheel increased by 280 rad s⁻¹ during the application of the angular impulse.

Determine the moment of inertia of the flywheel.

Calculate, using your answer to (b)(i), the time taken by the propellor to attain this rotational speed.

Calculate, using your answer to (b)(i), the time taken by the propellor to attain this rotational speed.

Calculate, using your answer to (b)(i), the time taken by the propellor to attain this rotational speed.

Calculate the angular impulse applied to the flywheel.

Calculate the angular impulse applied to the flywheel.

Calculate the angular impulse applied to the flywheel.

The flywheel was rotating at 150 rev per minute before the application of the angular impulse. Determine the change in angular rotational energy of the flywheel during the application of the flywheel.

The flywheel was rotating at 150 rev per minute before the application of the angular impulse. Determine the change in angular rotational energy of the flywheel during the application of the flywheel.

The flywheel was rotating at 150 rev per minute before the application of the angular impulse. Determine the change in angular rotational energy of the flywheel during the application of the flywheel.

the angular deceleration of the rod.

the angular deceleration of the rod.

the angular deceleration of the rod.

the angular deceleration of the rod.

the number of revolutions made by the rod until it stops rotating.

the number of revolutions made by the rod until it stops rotating.

the number of revolutions made by the rod until it stops rotating.

the number of revolutions made by the rod until it stops rotating.

Calculate the energy lost during the collision.

Calculate the energy lost during the collision.

Calculate the energy lost during the collision.

Calculate the energy lost during the collision.

Show that the angular speed of the system immediately after the collision is about 5.7 rad s⁻¹.

Show that the angular speed of the system immediately after the collision is about 5.7 rad s⁻¹.

Show that the angular speed of the system immediately after the collision is about 5.7 rad s⁻¹.

Show that the angular speed of the system immediately after the collision is about 5.7 rad s⁻¹.

Outline why the angular speed ω decreases when the spheres move outward.

Outline why the angular speed ω decreases when the spheres move outward.

Outline why the angular speed ω decreases when the spheres move outward.

Outline why the angular speed ω decreases when the spheres move outward.

Show that the rotational kinetic energy is $\frac{1}{2}$Lω where L is the angular momentum of the system.

Show that the rotational kinetic energy is $\frac{1}{2}$Lω where L is the angular momentum of the system.

Show that the rotational kinetic energy is $\frac{1}{2}$Lω where L is the angular momentum of the system.

Show that the rotational kinetic energy is $\frac{1}{2}$Lω where L is the angular momentum of the system.

Show that the net torque on the system about the central axis is approximately 30 N m.

Show that the net torque on the system about the central axis is approximately 30 N m.

Show that the net torque on the system about the central axis is approximately 30 N m.

Show that the net torque on the system about the central axis is approximately 30 N m.

The system rotates from rest and reaches a maximum angular speed of 20 rad s⁻¹ in a time of 5.0 s. Calculate the angular acceleration of the system.

The system rotates from rest and reaches a maximum angular speed of 20 rad s⁻¹ in a time of 5.0 s. Calculate the angular acceleration of the system.

The system rotates from rest and reaches a maximum angular speed of 20 rad s⁻¹ in a time of 5.0 s. Calculate the angular acceleration of the system.

The system rotates from rest and reaches a maximum angular speed of 20 rad s⁻¹ in a time of 5.0 s. Calculate the angular acceleration of the system.

Determine the moment of inertia of the system about the central axis.

Determine the moment of inertia of the system about the central axis.

Determine the moment of inertia of the system about the central axis.

Determine the moment of inertia of the system about the central axis.

When the spheres move outward, the angular speed decreases from 20 rad s⁻¹ to 12 rad s⁻¹. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward.

When the spheres move outward, the angular speed decreases from 20 rad s⁻¹ to 12 rad s⁻¹. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward.

When the spheres move outward, the angular speed decreases from 20 rad s⁻¹ to 12 rad s⁻¹. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward.

When the spheres move outward, the angular speed decreases from 20 rad s⁻¹ to 12 rad s⁻¹. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward.

Outline one reason why this model of a dancer is unrealistic.

Outline one reason why this model of a dancer is unrealistic.

Outline one reason why this model of a dancer is unrealistic.

Outline one reason why this model of a dancer is unrealistic.

A person stands in an elevator (lift). The total mass of the person and the elevator is 800 kg. The elevator accelerates upward at 2.0 m s⁻².

What is the tension in the cable?

A.  1.6 kN

B.  6.4 kN

C.  8.0 kN

D.  9.6 kN

A person stands in an elevator (lift). The total mass of the person and the elevator is 800 kg. The elevator accelerates upward at 2.0 m s⁻².

What is the tension in the cable?

A.  1.6 kN

B.  6.4 kN

C.  8.0 kN

D.  9.6 kN

A person stands in an elevator (lift). The total mass of the person and the elevator is 800 kg. The elevator accelerates upward at 2.0 m s⁻².

What is the tension in the cable?

A.  1.6 kN

B.  6.4 kN

C.  8.0 kN

D.  9.6 kN

A person stands in an elevator (lift). The total mass of the person and the elevator is 800 kg. The elevator accelerates upward at 2.0 m s⁻².

What is the tension in the cable?

A.  1.6 kN

B.  6.4 kN

C.  8.0 kN

D.  9.6 kN

A person stands in an elevator (lift). The total mass of the person and the elevator is 800 kg. The elevator accelerates upward at 2.0 m s⁻².

What is the tension in the cable?

A.  1.6 kN

B.  6.4 kN

C.  8.0 kN

D.  9.6 kN

A person stands in an elevator (lift). The total mass of the person and the elevator is 800 kg. The elevator accelerates upward at 2.0 m s⁻².

What is the tension in the cable?

A.  1.6 kN

B.  6.4 kN

C.  8.0 kN

D.  9.6 kN

A disc of mass M and radius R is on a horizontal frictionless table. Two equal and opposite forces, each of magnitude F, act on the disc. The moment of inertia of the disc about its axis is $\frac{1}{2}M{R}^2$.

What is the angular acceleration of the disc?

A.  0

B.  $\frac{F}{MR}$

C.  $\frac{2F}{MR}$

D. $\frac{4F}{MR}$

A disc of mass M and radius R is on a horizontal frictionless table. Two equal and opposite forces, each of magnitude F, act on the disc. The moment of inertia of the disc about its axis is $\frac{1}{2}M{R}^2$.

What is the angular acceleration of the disc?

A.  0

B.  $\frac{F}{MR}$

C.  $\frac{2F}{MR}$

D. $\frac{4F}{MR}$

A disc of mass M and radius R is on a horizontal frictionless table. Two equal and opposite forces, each of magnitude F, act on the disc. The moment of inertia of the disc about its axis is $\frac{1}{2}M{R}^2$.

What is the angular acceleration of the disc?

A.  0

B.  $\frac{F}{MR}$

C.  $\frac{2F}{MR}$

D. $\frac{4F}{MR}$

A disc of mass M and radius R is on a horizontal frictionless table. Two equal and opposite forces, each of magnitude F, act on the disc. The moment of inertia of the disc about its axis is $\frac{1}{2}M{R}^2$.

What is the angular acceleration of the disc?

A.  0

B.  $\frac{F}{MR}$

C.  $\frac{2F}{MR}$

D. $\frac{4F}{MR}$

A disc of mass M and radius R is on a horizontal frictionless table. Two equal and opposite forces, each of magnitude F, act on the disc. The moment of inertia of the disc about its axis is $\frac{1}{2}M{R}^2$.

What is the angular acceleration of the disc?

A.  0

B.  $\frac{F}{MR}$

C.  $\frac{2F}{MR}$

D. $\frac{4F}{MR}$

A disc of mass M and radius R is on a horizontal frictionless table. Two equal and opposite forces, each of magnitude F, act on the disc. The moment of inertia of the disc about its axis is $\frac{1}{2}M{R}^2$.

What is the angular acceleration of the disc?

A.  0

B.  $\frac{F}{MR}$

C.  $\frac{2F}{MR}$

D. $\frac{4F}{MR}$

A cylinder of mass $M$ and radius $R$ rotates at constant angular speed ω about an axis through its centre. The rotational kinetic energy of the cylinder is K.

The moment of inertia of the cylinder is $\frac{1}{2}M{R}^2$.

A second cylinder has mass $2M$, radius $2R$ and rotates with angular speed 2ω.

What is the rotational kinetic energy of the second cylinder?

A.  8K

B.  16K

C.  32K

D.  64K

A cylinder of mass $M$ and radius $R$ rotates at constant angular speed ω about an axis through its centre. The rotational kinetic energy of the cylinder is K.

The moment of inertia of the cylinder is $\frac{1}{2}M{R}^2$.

A second cylinder has mass $2M$, radius $2R$ and rotates with angular speed 2ω.

What is the rotational kinetic energy of the second cylinder?

A.  8K

B.  16K

C.  32K

D.  64K

A cylinder of mass $M$ and radius $R$ rotates at constant angular speed ω about an axis through its centre. The rotational kinetic energy of the cylinder is K.

The moment of inertia of the cylinder is $\frac{1}{2}M{R}^2$.

A second cylinder has mass $2M$, radius $2R$ and rotates with angular speed 2ω.

What is the rotational kinetic energy of the second cylinder?

A.  8K

B.  16K

C.  32K

D.  64K

A cylinder of mass $M$ and radius $R$ rotates at constant angular speed ω about an axis through its centre. The rotational kinetic energy of the cylinder is K.

The moment of inertia of the cylinder is $\frac{1}{2}M{R}^2$.

A second cylinder has mass $2M$, radius $2R$ and rotates with angular speed 2ω.

What is the rotational kinetic energy of the second cylinder?

A.  8K

B.  16K

C.  32K

D.  64K

Show that the angular acceleration $a$ of the cylinder is $\frac{g}{3R}$

Show that the angular acceleration $a$ of the cylinder is $\frac{g}{3R}$

Show that the tension T in the string is$\frac{Mg}{6}$

Show that the tension T in the string is$\frac{Mg}{6}$

The block falls a distance 0.50 m after its release before hitting the ground. Show that the block hits the ground 0.55 s after release.

The block falls a distance 0.50 m after its release before hitting the ground. Show that the block hits the ground 0.55 s after release.

Calculate, for the cylinder, at the instant just before the block hits the ground the angular momentum.

Calculate, for the cylinder, at the instant just before the block hits the ground the angular momentum.

Calculate, for the cylinder, at the instant just before the block hits the ground the kinetic energy.

Calculate, for the cylinder, at the instant just before the block hits the ground the kinetic energy.

Calculate F.

Calculate F.

Show that the total kinetic energy E_k of the sphere when it rolls, without slipping, at speed v is $E_{\text{K}}=\frac{7}{10}m{v}^2$.

Show that the total kinetic energy E_k of the sphere when it rolls, without slipping, at speed v is $E_{\text{K}}=\frac{7}{10}m{v}^2$.

A solid sphere of mass 1.5 kg is rolling, without slipping, on a horizontal surface with a speed of 0.50 m s^-1. The sphere then rolls, without slipping, down a ramp to reach a horizontal surface that is 45 cm lower.

Calculate the speed of the sphere at the bottom of the ramp.

A solid sphere of mass 1.5 kg is rolling, without slipping, on a horizontal surface with a speed of 0.50 m s^-1. The sphere then rolls, without slipping, down a ramp to reach a horizontal surface that is 45 cm lower.

Calculate the speed of the sphere at the bottom of the ramp.

Show that the pressure at B is about 130 kPa.

Show that the pressure at B is about 130 kPa.

Calculate the ratio $\frac{V_{\mathrm{A}}}{V_{\mathrm{C}}}$.

Calculate the ratio $\frac{V_{\mathrm{A}}}{V_{\mathrm{C}}}$.

determine the thermal energy removed from the system.

determine the thermal energy removed from the system.

explain why the entropy of the gas decreases.

explain why the entropy of the gas decreases.

Show that the final angular velocity of the bar is about $3 \mathrm{rad} {\mathrm{s}}^{-1}$.

Show that the final angular velocity of the bar is about $3 \mathrm{rad} {\mathrm{s}}^{-1}$.

Draw the variation with time $t$ of the angular displacement $\theta$ of the bar during the acceleration.

Draw the variation with time $t$ of the angular displacement $\theta$ of the bar during the acceleration.

Calculate the torque acting on the bar while it is accelerating.

Calculate the torque acting on the bar while it is accelerating.

The torque is removed. The bar comes to rest in $30$ complete rotations with constant angular deceleration. Determine the time taken for the bar to come to rest.

The torque is removed. The bar comes to rest in $30$ complete rotations with constant angular deceleration. Determine the time taken for the bar to come to rest.

Explain the direction in which the person-turntable system starts to rotate.

Explain the direction in which the person-turntable system starts to rotate.

A solid sphere of radius $r$ and mass $m$ is released from rest and rolls down a slope, without slipping. The vertical height of the slope is $h$. The moment of inertia $I$ of this sphere about an axis through its centre is $\frac{2}{5}m{r}^2$.

Show that the linear velocity $v$ of the sphere as it leaves the slope is $\sqrt{\frac{10gh}{7}}$.

A solid sphere of radius $r$ and mass $m$ is released from rest and rolls down a slope, without slipping. The vertical height of the slope is $h$. The moment of inertia $I$ of this sphere about an axis through its centre is $\frac{2}{5}m{r}^2$.

Show that the linear velocity $v$ of the sphere as it leaves the slope is $\sqrt{\frac{10gh}{7}}$.

Show that the angular speed of the system immediately after the collision is about 5.7 rad s⁻¹.

Show that the angular speed of the system immediately after the collision is about 5.7 rad s⁻¹.

Show that the angular speed of the system immediately after the collision is about 5.7 rad s⁻¹.

Show that the angular speed of the system immediately after the collision is about 5.7 rad s⁻¹.

Calculate the energy lost during the collision.

Calculate the energy lost during the collision.

Calculate the energy lost during the collision.

Calculate the energy lost during the collision.

the angular deceleration of the rod.

the angular deceleration of the rod.

the angular deceleration of the rod.

the angular deceleration of the rod.

the number of revolutions made by the rod until it stops rotating.

the number of revolutions made by the rod until it stops rotating.

the number of revolutions made by the rod until it stops rotating.

the number of revolutions made by the rod until it stops rotating.

Show that the net torque on the system about the central axis is approximately 30 N m.

Show that the net torque on the system about the central axis is approximately 30 N m.

Show that the net torque on the system about the central axis is approximately 30 N m.

Show that the net torque on the system about the central axis is approximately 30 N m.

The system rotates from rest and reaches a maximum angular speed of 20 rad s⁻¹ in a time of 5.0 s. Calculate the angular acceleration of the system.

The system rotates from rest and reaches a maximum angular speed of 20 rad s⁻¹ in a time of 5.0 s. Calculate the angular acceleration of the system.

The system rotates from rest and reaches a maximum angular speed of 20 rad s⁻¹ in a time of 5.0 s. Calculate the angular acceleration of the system.

The system rotates from rest and reaches a maximum angular speed of 20 rad s⁻¹ in a time of 5.0 s. Calculate the angular acceleration of the system.

Determine the moment of inertia of the system about the central axis.

Determine the moment of inertia of the system about the central axis.

Determine the moment of inertia of the system about the central axis.

Determine the moment of inertia of the system about the central axis.

Outline why the angular speed ω decreases when the spheres move outward.

Outline why the angular speed ω decreases when the spheres move outward.

Outline why the angular speed ω decreases when the spheres move outward.

Outline why the angular speed ω decreases when the spheres move outward.

Show that the rotational kinetic energy is $\frac{1}{2}$Lω where L is the angular momentum of the system.

Show that the rotational kinetic energy is $\frac{1}{2}$Lω where L is the angular momentum of the system.

Show that the rotational kinetic energy is $\frac{1}{2}$Lω where L is the angular momentum of the system.

Show that the rotational kinetic energy is $\frac{1}{2}$Lω where L is the angular momentum of the system.

When the spheres move outward, the angular speed decreases from 20 rad s⁻¹ to 12 rad s⁻¹. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward.

When the spheres move outward, the angular speed decreases from 20 rad s⁻¹ to 12 rad s⁻¹. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward.

When the spheres move outward, the angular speed decreases from 20 rad s⁻¹ to 12 rad s⁻¹. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward.

When the spheres move outward, the angular speed decreases from 20 rad s⁻¹ to 12 rad s⁻¹. Calculate the percentage change in rotational kinetic energy that occurs when the spheres move outward.

Outline one reason why this model of a dancer is unrealistic.

Outline one reason why this model of a dancer is unrealistic.

Outline one reason why this model of a dancer is unrealistic.

Outline one reason why this model of a dancer is unrealistic.