A 2 kg mass moving at 18 m/s strikes a stationary 8 kg mass and they move together. Find the common velocity and the energy lost.

Start with the thinking

• Momentum is conserved in every impact; energy is not.
• Plastic impact means one common final velocity.

Step-by-step solution

1. Momentum

   $$m_{1}v_{1} + m_{2}v_{2} = (m_{1} + m_{2})v'$$

2. Substituting

   $$2(18) + 8(0) = (10)v'$$

3. Solve

   $$v' = 36/10 = 3.600 m/s$$

4. Initial KE

   $$½(2)(18)^{2} = 324.0 J$$

5. Final KE

   $$½(10)(3.600)^{2} = 64.8 J$$

6. Energy lost — ΔKE = 259.2 J

Why the other options are there

• 9.00 m/s (masses assumed equal)
• ΔKE = 0 (energy assumed conserved)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum

A drum with mass moment of inertia 21.0 kg·m² is driven by a constant torque of 132 N·m from rest. Find α and the angular speed after 5.5 s.

Start with the thinking

• Rotational analogue of F = ma is M = Iα.
• Constant torque gives constant angular acceleration.

Step-by-step solution

1. Rotational equation

   $$M = I\alpha$$

2. Substituting

   $$\alpha = 132/21.0 = 6.286 rad/s^{2}$$

3. Angular speed — ω = ω₀ + αt

4. Substituting

   $$\omega = 0 + 6.286(5.5) = 34.57 rad/s$$

5. Convert

   $$\omega = 330.1 rpm$$

Why the other options are there

• 2,772 rad/s² (multiplied instead of divided)
• 5.50 rad/s (revolutions confused with radians)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum

A 8 kg mass moving at 16 m/s strikes a stationary 9 kg mass and they move together. Find the common velocity and the energy lost.

Start with the thinking

• Momentum is conserved in every impact; energy is not.
• Plastic impact means one common final velocity.

Step-by-step solution

1. Momentum

   $$m_{1}v_{1} + m_{2}v_{2} = (m_{1} + m_{2})v'$$

2. Substituting

   $$8(16) + 9(0) = (17)v'$$

3. Solve

   $$v' = 128/17 = 7.529 m/s$$

4. Initial KE

   $$½(8)(16)^{2} = 1,024 J$$

5. Final KE

   $$½(17)(7.529)^{2} = 481.9 J$$

6. Energy lost — ΔKE = 542.1 J

Why the other options are there

• 8.00 m/s (masses assumed equal)
• ΔKE = 0 (energy assumed conserved)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum

A drum with mass moment of inertia 23.5 kg·m² is driven by a constant torque of 31 N·m from rest. Find α and the angular speed after 4.0 s.

Start with the thinking

• Rotational analogue of F = ma is M = Iα.
• Constant torque gives constant angular acceleration.

Step-by-step solution

1. Rotational equation

   $$M = I\alpha$$

2. Substituting

   $$\alpha = 31/23.5 = 1.319 rad/s^{2}$$

3. Angular speed — ω = ω₀ + αt

4. Substituting

   $$\omega = 0 + 1.319(4.0) = 5.28 rad/s$$

5. Convert

   $$\omega = 50.4 rpm$$

Why the other options are there

• 728.5 rad/s² (multiplied instead of divided)
• 0.84 rad/s (revolutions confused with radians)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum

A 4 kg mass moving at 7 m/s strikes a stationary 8 kg mass and they move together. Find the common velocity and the energy lost.

Start with the thinking

• Momentum is conserved in every impact; energy is not.
• Plastic impact means one common final velocity.

Step-by-step solution

1. Momentum

   $$m_{1}v_{1} + m_{2}v_{2} = (m_{1} + m_{2})v'$$

2. Substituting

   $$4(7) + 8(0) = (12)v'$$

3. Solve

   $$v' = 28/12 = 2.333 m/s$$

4. Initial KE

   $$½(4)(7)^{2} = 98.0 J$$

5. Final KE

   $$½(12)(2.333)^{2} = 32.7 J$$

6. Energy lost — ΔKE = 65.3 J

Why the other options are there

• 3.50 m/s (masses assumed equal)
• ΔKE = 0 (energy assumed conserved)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum

A drum with mass moment of inertia 21.5 kg·m² is driven by a constant torque of 99 N·m from rest. Find α and the angular speed after 5.0 s.

Start with the thinking

• Rotational analogue of F = ma is M = Iα.
• Constant torque gives constant angular acceleration.

Step-by-step solution

1. Rotational equation

   $$M = I\alpha$$

2. Substituting

   $$\alpha = 99/21.5 = 4.605 rad/s^{2}$$

3. Angular speed — ω = ω₀ + αt

4. Substituting

   $$\omega = 0 + 4.605(5.0) = 23.02 rad/s$$

5. Convert

   $$\omega = 219.9 rpm$$

Why the other options are there

• 2,129 rad/s² (multiplied instead of divided)
• 3.66 rad/s (revolutions confused with radians)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum

A 3 kg mass moving at 11 m/s strikes a stationary 10 kg mass and they move together. Find the common velocity and the energy lost.

Start with the thinking

• Momentum is conserved in every impact; energy is not.
• Plastic impact means one common final velocity.

Step-by-step solution

1. Momentum

   $$m_{1}v_{1} + m_{2}v_{2} = (m_{1} + m_{2})v'$$

2. Substituting

   $$3(11) + 10(0) = (13)v'$$

3. Solve

   $$v' = 33/13 = 2.538 m/s$$

4. Initial KE

   $$½(3)(11)^{2} = 181.5 J$$

5. Final KE

   $$½(13)(2.538)^{2} = 41.9 J$$

6. Energy lost — ΔKE = 139.6 J

Why the other options are there

• 5.50 m/s (masses assumed equal)
• ΔKE = 0 (energy assumed conserved)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum

A drum with mass moment of inertia 30.0 kg·m² is driven by a constant torque of 170 N·m from rest. Find α and the angular speed after 5.5 s.

Start with the thinking

• Rotational analogue of F = ma is M = Iα.
• Constant torque gives constant angular acceleration.

Step-by-step solution

1. Rotational equation

   $$M = I\alpha$$

2. Substituting

   $$\alpha = 170/30.0 = 5.667 rad/s^{2}$$

3. Angular speed — ω = ω₀ + αt

4. Substituting

   $$\omega = 0 + 5.667(5.5) = 31.17 rad/s$$

5. Convert

   $$\omega = 297.6 rpm$$

Why the other options are there

• 5,100 rad/s² (multiplied instead of divided)
• 4.96 rad/s (revolutions confused with radians)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum

A 6 kg mass moving at 15 m/s strikes a stationary 7 kg mass and they move together. Find the common velocity and the energy lost.

Start with the thinking

• Momentum is conserved in every impact; energy is not.
• Plastic impact means one common final velocity.

Step-by-step solution

1. Momentum

   $$m_{1}v_{1} + m_{2}v_{2} = (m_{1} + m_{2})v'$$

2. Substituting

   $$6(15) + 7(0) = (13)v'$$

3. Solve

   $$v' = 90/13 = 6.923 m/s$$

4. Initial KE

   $$½(6)(15)^{2} = 675.0 J$$

5. Final KE

   $$½(13)(6.923)^{2} = 311.5 J$$

6. Energy lost — ΔKE = 363.5 J

Why the other options are there

• 7.50 m/s (masses assumed equal)
• ΔKE = 0 (energy assumed conserved)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum

A drum with mass moment of inertia 23.5 kg·m² is driven by a constant torque of 69 N·m from rest. Find α and the angular speed after 9.5 s.

Start with the thinking

• Rotational analogue of F = ma is M = Iα.
• Constant torque gives constant angular acceleration.

Step-by-step solution

1. Rotational equation

   $$M = I\alpha$$

2. Substituting

   $$\alpha = 69/23.5 = 2.936 rad/s^{2}$$

3. Angular speed — ω = ω₀ + αt

4. Substituting

   $$\omega = 0 + 2.936(9.5) = 27.89 rad/s$$

5. Convert

   $$\omega = 266.4 rpm$$

Why the other options are there

• 1,622 rad/s² (multiplied instead of divided)
• 4.44 rad/s (revolutions confused with radians)

Reference: FE Reference Handbook — Dynamics → Principle of Angular Impulse and Momentum