Physics 10310 Final Exam Study Guidance

Q1. An object has an initial momentum $\vec{p}_i = b\hat{i}$. Starting at $t = 0$, a force $\vec{F}(t) = ct\hat{i} + dt^2\hat{j}$ is applied. What is the magnitude of the object's momentum at $t = t_f$? Express your answer in terms of $b, c, d, t_f$ and any needed constants.

Background

Topic: Impulse and Momentum in Two Dimensions

This question tests your understanding of how impulse (the integral of force over time) changes an object's momentum, and how to find the magnitude of the resulting vector.

Key Terms and Formulas:

• Impulse: $\vec{J} = \int_{t_1}^{t_2} \vec{F}(t) dt$

• Momentum Update: $\vec{p}_f = \vec{p}_i + \vec{J}$

• Magnitude of a vector: $|\vec{p}_f| = \sqrt{(p_{fx})^2 + (p_{fy})^2}$

Step-by-Step Guidance

1. Write the impulse as the integral of the force from $t = 0$ to $t = t_f$:

   $\vec{J} = \int_0^{t_f} (ct\hat{i} + dt^2\hat{j}) dt$

2. Integrate each component separately to find $J_x$ and $J_y$:

   $J_x = \int_0^{t_f} ct \, dt$

   $J_y = \int_0^{t_f} dt^2 \, dt$

3. Calculate the new momentum components by adding the impulse to the initial momentum:

   $p_{fx} = p_{ix} + J_x$

   $p_{fy} = p_{iy} + J_y$

   Recall that $p_{ix} = b$ and $p_{iy} = 0$.

4. Write the final momentum vector in component form:

   $\vec{p}_f = (b + J_x)\hat{i} + (J_y)\hat{j}$

5. Set up the expression for the magnitude of the final momentum:

   $|\vec{p}_f| = \sqrt{(b + J_x)^2 + (J_y)^2}$

Try solving on your own before revealing the answer!

Final Answer:

$J_x = \frac{c}{2} t_f^2$ and $J_y = \frac{d}{3} t_f^3$

The magnitude is $|\vec{p}_f| = \sqrt{\left(b + \frac{c}{2} t_f^2\right)^2 + \left(\frac{d}{3} t_f^3\right)^2}$

This uses the impulse-momentum theorem and vector magnitude formula.

Q2. On the surface of Phobos (radius $R$, mass $M$), how fast must you throw a baseball horizontally for it to orbit? Express your answer in terms of $M, R$, and any necessary constants.

Background

Topic: Circular Motion and Universal Gravitation

This question tests your ability to relate gravitational force to the required centripetal force for circular motion just above a planet's surface.

Key Terms and Formulas:

• Gravitational Force: $F_g = \frac{GMm}{R^2}$

• Centripetal Force: $F_c = \frac{mv^2}{R}$

• Set $F_g = F_c$ for a stable orbit at the surface.

Step-by-Step Guidance

1. Write the gravitational force acting on the baseball at the surface:

   $F_g = \frac{GMm}{R^2}$

2. Write the required centripetal force for circular motion:

   $F_c = \frac{mv^2}{R}$

3. Set the gravitational force equal to the centripetal force:

   $\frac{GMm}{R^2} = \frac{mv^2}{R}$

4. Solve for $v$ in terms of $G, M, R$.

Try solving on your own before revealing the answer!

Final Answer:

$v = \sqrt{\frac{GM}{R}}$

This is the minimum speed for a circular orbit just above the surface, derived by equating gravitational and centripetal forces.

Q3. A uniform rod (mass $M$, length $L$) rotates about its center. Two beads (mass $m$ each) are mounted a distance $\theta$ from the axis. The rod rotates with angular velocity $\omega_i$. When the beads are released, they slide to the ends. What is the rod's angular velocity when the beads reach the ends?

Background

Topic: Conservation of Angular Momentum

This question tests your understanding of how the moment of inertia changes and how angular momentum is conserved when mass redistributes.

Key Terms and Formulas:

• Moment of inertia for a rod about its center: $I_{rod} = \frac{1}{12}ML^2$

• Moment of inertia for a point mass: $I = mr^2$

• Angular momentum conservation: $I_i \omega_i = I_f \omega_f$

Step-by-Step Guidance

1. Write the initial moment of inertia:

   $I_i = \frac{1}{12}ML^2 + 2m\theta^2$

2. Write the final moment of inertia (beads at the ends):

   $I_f = \frac{1}{12}ML^2 + 2m\left(\frac{L}{2}\right)^2$

3. Set up angular momentum conservation:

   $I_i \omega_i = I_f \omega_f$

4. Rearrange to solve for $\omega_f$ in terms of $M, L, m, \theta, \omega_i$.

Try solving on your own before revealing the answer!

Final Answer:

$\omega_f = \omega_i \frac{\frac{1}{12}ML^2 + 2m\theta^2}{\frac{1}{12}ML^2 + 2m\left(\frac{L}{2}\right)^2}$

This uses conservation of angular momentum as the system is isolated and no external torques act.

Q4. A solid ball (radius $R$) rolls without slipping down a ramp from rest. It leaves the ramp from height $h$ at angle $\varphi$ to the horizontal and lands after time $t$. From what initial height (??) was the ball released? Express in terms of $R, h, t, \varphi$, and constants.

Background

Topic: Energy Conservation and Projectile Motion

This question tests your ability to use energy conservation (including rotational kinetic energy) and kinematics to relate the initial height to the projectile's motion.

Key Terms and Formulas:

• Energy conservation: $m g H = m g h + \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$

• Moment of inertia for a solid sphere: $I = \frac{2}{5} m R^2$

• Rolling without slipping: $v = \omega R$

• Projectile kinematics: $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

Step-by-Step Guidance

1. Apply energy conservation from the initial height (??) to the point where the ball leaves the ramp:

   $m g H = m g h + \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$

2. Substitute $I$ and use $v = \omega R$ to combine kinetic energy terms.

3. Solve for $v$ in terms of $H, h, R$ and constants.

4. Use projectile motion equations to relate $v$ and $t$ to the vertical displacement ($h$ to ground) and angle $\varphi$.

5. Set up the equation for $H$ (the initial height) in terms of $h, t, \varphi, R$, and $g$.

Try solving on your own before revealing the answer!

Final Answer:

$H = h + \frac{v^2}{2g} \left(1 + \frac{2}{5}\right)$, with $v$ found from kinematics: $v = \frac{(h + \frac{1}{2}gt^2)}{t \sin \varphi}$

Combine energy and kinematics to solve for the initial height.

Q5. A ladder rests between a glass wall (no friction) and a grassy lawn (static friction coefficient $\mu_s$). What is the smallest angle the ladder can make with the ground without slipping? Express in terms of $\mu_s$ and constants.

Background

Topic: Static Equilibrium and Friction

This question tests your ability to analyze forces and torques for equilibrium, including friction and normal forces.

Key Terms and Formulas:

• Sum of forces in $x$ and $y$: $\sum F_x = 0$, $\sum F_y = 0$

• Sum of torques: $\sum \tau = 0$

• Friction force: $f_s \leq \mu_s N$

Step-by-Step Guidance

1. Draw a free-body diagram showing all forces: gravity, normal forces at wall and ground, friction at ground.

2. Write equilibrium equations for forces in $x$ and $y$ directions.

3. Write the torque equilibrium equation about the base of the ladder.

4. Express the maximum static friction force in terms of $\mu_s$ and the normal force.

5. Combine the equations to solve for the minimum angle $\theta$ in terms of $\mu_s$.

Try solving on your own before revealing the answer!

Final Answer:

$\tan \theta_{min} = 2 \mu_s$

This comes from balancing torques and using the maximum static friction condition.