College Physics I: Step-by-Step Guidance for Final Exam Review Problems (Chapters 7–9)

Q1A. During a collision with a wall, the velocity of a 0.200-kg ball changes from 20.0 m/s toward the wall to 12.0 m/s away from the wall. If the time the ball was in contact with the wall was 60.0 ms, what was the magnitude of the average force applied to the ball?

Background

Topic: Impulse and Momentum

This question tests your understanding of the impulse-momentum theorem, which relates the change in momentum of an object to the average force applied over a time interval.

Key Terms and Formulas

• Impulse ($J$): The product of average force and the time interval during which the force acts.

• Momentum ($p$): The product of mass and velocity.

• Impulse-Momentum Theorem: $J = \Delta p = F_{avg} \Delta t$

Step-by-Step Guidance

1. Calculate the initial and final momentum of the ball. Remember to consider the direction of velocity (toward and away from the wall).

   $p_{initial} = m v_{initial}$

   $p_{final} = m v_{final}$

2. Find the change in momentum ($\Delta p$). Be careful with the signs, since the direction changes.

   $\Delta p = p_{final} - p_{initial}$

3. Use the impulse-momentum theorem to relate the change in momentum to the average force and the time interval.

   $F_{avg} = \frac{\Delta p}{\Delta t}$

4. Convert the time from milliseconds to seconds before substituting values.

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Q1B. An object, initially at rest, explodes into two fragments. The first, of mass 14 g, moves in the +x-direction at 48 m/s. The second moves at 32 m/s. What are the mass and direction of the second fragment?

Background

Topic: Conservation of Momentum (Explosions)

This question tests your ability to apply conservation of momentum to a system where an object at rest explodes into two fragments moving in opposite directions.

Key Terms and Formulas

• Conservation of Momentum: $\sum p_{initial} = \sum p_{final}$

• Momentum: $p = m v$

Step-by-Step Guidance

1. Write the equation for conservation of momentum. Since the object is initially at rest, the total initial momentum is zero.

   $0 = m_1 v_1 + m_2 v_2$

2. Assign directions: Let the +x direction be positive. If $v_1$ is positive, $v_2$ must be negative (opposite direction) for momentum to be conserved.

3. Plug in the known values for $m_1$, $v_1$, and $v_2$ (be careful with the sign for $v_2$).

4. Solve for the unknown mass $m_2$ and determine the direction based on the sign of $v_2$.

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Q2A. A 14,000-kg boxcar is coasting at 1.50 m/s along a horizontal track when it suddenly hits and couples with a stationary 10,000-kg boxcar. What is the speed of the cars just after the collision?

Background

Topic: Conservation of Momentum (Inelastic Collisions)

This question tests your ability to apply conservation of momentum to a perfectly inelastic collision, where two objects stick together after colliding.

Key Terms and Formulas

• Conservation of Momentum: $m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_{final}$

Step-by-Step Guidance

1. Identify the masses and velocities of both boxcars before the collision.

   $m_1 = 14,000$ kg, $v_1 = 1.50$ m/s; $m_2 = 10,000$ kg, $v_2 = 0$ m/s

2. Write the conservation of momentum equation for the system.

   $m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_{final}$

3. Plug in the known values and solve for $v_{final}$.

   $v_{final} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$

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Q2B. A 1000-kg car traveling west at 10.0 m/s collides with a 500-kg car traveling south at 20.0 m/s. The cars stick together. (a) What is the speed of the wreckage just after the collision? (b) In what direction does the wreckage move just after the collision?

Background

Topic: Conservation of Momentum in Two Dimensions

This question tests your ability to apply conservation of momentum in two perpendicular directions and to find the magnitude and direction of the resulting velocity vector.

Key Terms and Formulas

• Momentum in x-direction: $p_x = m v_x$

• Momentum in y-direction: $p_y = m v_y$

• Resultant speed: $v = \sqrt{v_x^2 + v_y^2}$

• Direction: $\theta = \arctan\left(\frac{v_y}{v_x}\right)$

Step-by-Step Guidance

1. Calculate the total momentum in the x-direction (west) and y-direction (south) before the collision.

2. Since the cars stick together, add the momenta vectorially to get the total momentum after the collision.

3. Find the magnitude of the velocity of the combined wreckage using the Pythagorean theorem.

   $v = \frac{\sqrt{(p_x)^2 + (p_y)^2}}{m_{total}}$

4. Find the direction of motion using the arctangent function.

   $\theta = \arctan\left(\frac{p_y}{p_x}\right)$

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Q3. Determine the net torque on the 2.0-m-long uniform beam shown in the figure below. All forces are shown. Calculate about (a) point C, the CM, and (b) point P at one end.

Background

Topic: Torque and Rotational Equilibrium

This question tests your ability to calculate net torque about different points on a beam, using the definition of torque and considering the lever arm and force directions.

Key Terms and Formulas

• Torque: $\tau = r F \sin \theta$

• Net torque: $\sum \tau = \sum (r_i F_i \sin \theta_i)$

Step-by-Step Guidance

1. Identify all forces acting on the beam and their points of application (distances from the chosen axis).

2. For each force, calculate the torque about the specified point (C or P), using the perpendicular distance from the axis to the line of action of the force.

3. Assign the correct sign to each torque (positive for counterclockwise, negative for clockwise).

4. Sum all the torques to find the net torque about the chosen point.

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Q4. A uniform disk of mass M and radius R is mounted on a fixed horizontal axle. A block with mass m hangs from a massless cord that is wrapped around the rim of the disk. Find the acceleration of the block, angular acceleration of the disk, and the tension in the cord. The cord does not slip, and there is no friction at the axle.

Background

Topic: Rotational Dynamics (Atwood's Machine Variant)

This question tests your understanding of Newton's second law for rotation, the relationship between linear and angular acceleration, and the forces acting on a system involving both rotation and translation.

Key Terms and Formulas

• Newton's Second Law (linear): $m a = m g - T$

• Newton's Second Law (rotational): $I \alpha = T R$

• Moment of inertia for a disk: $I = \frac{1}{2} M R^2$

• Relationship between linear and angular acceleration: $a = \alpha R$

Step-by-Step Guidance

1. Draw a free-body diagram for the block and the disk. Identify all forces and torques.

2. Write Newton's second law for the block (vertical direction): $m a = m g - T$

3. Write Newton's second law for rotation for the disk: $I \alpha = T R$

4. Relate the linear acceleration of the block to the angular acceleration of the disk: $a = \alpha R$

5. Combine the equations to solve for acceleration $a$, angular acceleration $\alpha$, and tension $T$ in terms of $m$, $M$, $g$, and $R$.

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Q5. A potter’s wheel is a stone disk 90 cm in diameter with mass 120 kg. If the potter’s foot pushes at the outer edge of the initially stationary wheel with a 75-N force for one-eighth of a revolution, what will be the final speed? (Moment of inertia of a uniform disk: $I = \frac{1}{2} M r^2$)

Background

Topic: Work and Energy in Rotational Motion

This question tests your ability to relate work done by a force applied tangentially to a rotating disk to the resulting angular speed, using the work-energy theorem for rotation.

Key Terms and Formulas

• Work done by a force: $W = F \cdot d$ (where $d$ is the distance along the rim)

• Rotational kinetic energy: $K = \frac{1}{2} I \omega^2$

• Moment of inertia for a disk: $I = \frac{1}{2} M r^2$

• Relationship between work and change in kinetic energy: $W = \Delta K$

Step-by-Step Guidance

1. Calculate the distance along the rim for one-eighth of a revolution: $d = \frac{1}{8} \times 2 \pi r$

2. Calculate the work done: $W = F \cdot d$

3. Set the work done equal to the rotational kinetic energy: $W = \frac{1}{2} I \omega^2$

4. Solve for the final angular speed $\omega$.

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Q6. A 3.0-m-diameter merry-go-round with rotational inertia 120 kg·m² is spinning freely at 0.50 rev/s. Four 25-kg children sit suddenly on the edge of the merry-go-round. (a) Find the new angular speed, and (b) determine the change in kinetic energy.

Background

Topic: Conservation of Angular Momentum

This question tests your ability to apply conservation of angular momentum when the moment of inertia of a system changes, and to calculate the resulting change in kinetic energy.

Key Terms and Formulas

• Angular momentum: $L = I \omega$

• Conservation of angular momentum: $I_{initial} \omega_{initial} = I_{final} \omega_{final}$

• Rotational kinetic energy: $K = \frac{1}{2} I \omega^2$

Step-by-Step Guidance

1. Calculate the initial angular momentum using the initial moment of inertia and angular speed.

2. Find the new moment of inertia after the children sit on the edge (add their moments of inertia to the merry-go-round's).

3. Apply conservation of angular momentum to solve for the new angular speed.

4. Calculate the initial and final rotational kinetic energies, then find the change in kinetic energy.

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Q7. A skater has rotational inertia 4.2 kg·m² with his fists held to his chest and 5.7 kg·m² with his arms outstretched. The skater is spinning at 3.0 rev/s while holding a 2.5-kg weight in each outstretched hand; the weights are 76 cm from his rotation axis. If he pulls his hands in to his chest, so they’re essentially on his rotation axis, how fast will he be spinning?

Background

Topic: Conservation of Angular Momentum (Variable Moment of Inertia)

This question tests your understanding of how angular speed changes when the distribution of mass (moment of inertia) changes, while angular momentum is conserved.

Key Terms and Formulas

• Angular momentum: $L = I \omega$

• Conservation of angular momentum: $I_{initial} \omega_{initial} = I_{final} \omega_{final}$

Step-by-Step Guidance

1. Calculate the initial moment of inertia, including the skater and the weights (use $I = m r^2$ for each weight).

2. Calculate the final moment of inertia when the weights are at the axis (their contribution is zero).

3. Apply conservation of angular momentum to solve for the final angular speed.

   $I_{initial} \omega_{initial} = I_{final} \omega_{final}$

Try solving on your own before revealing the answer!

Q8. A 4.2-m-long beam is supported by a cable at its center. A 65-kg steelworker stands at one end of the beam. Where should a 190-kg bucket of concrete be suspended for the beam to be in static equilibrium?

Background

Topic: Static Equilibrium and Torque

This question tests your ability to set up the conditions for static equilibrium (sum of torques equals zero) and solve for the unknown position of a mass.

Key Terms and Formulas

• Torque: $\tau = r F$ (for forces perpendicular to the lever arm)

• Static equilibrium: $\sum \tau = 0$

Step-by-Step Guidance

1. Choose a pivot point (often the center of the beam, where the cable supports it).

2. Write the torque equation about the pivot, considering the steelworker and the bucket (and the beam's own weight if needed).

3. Set the sum of torques equal to zero and solve for the unknown distance from the center where the bucket should be suspended.

Try solving on your own before revealing the answer!