Physics Study Guide: Forces, Energy, and Rotational Motion

Q1. A spring with k = 120 N/m, length = 0.60 m, and mass = 0.40 kg is hung vertically. What is the extension when a mass of 0.50 kg is attached?

Background

Topic: Hooke's Law and Equilibrium

This question tests your understanding of Hooke's Law and how to calculate the extension of a spring when a mass is attached, considering equilibrium between the spring force and gravitational force.

Key Terms and Formulas

• Hooke's Law: $F = kx$

• Gravitational Force: $F_g = mg$

• Equilibrium: $F_{spring} = F_g$

• Extension: $x$ (distance spring stretches)

Step-by-Step Guidance

1. Identify the forces acting on the mass: the spring force upward and the gravitational force downward.

2. Set up the equilibrium equation: $k x = m g$

3. Rearrange to solve for the extension: $x = \frac{m g}{k}$

4. Plug in the values: $m = 0.50$ kg, $g = 9.8$ m/s$^2$, $k = 120$ N/m.

Try solving on your own before revealing the answer!

Final Answer: 0.041 m

$x = \frac{0.50 \times 9.8}{120} = 0.041$ m

The spring stretches by 0.041 meters when the mass is attached.

Q2. What is the acceleration of a 2 kg mass hanging from a spring with a spring constant of 50 N/m?

Background

Topic: Simple Harmonic Motion (SHM)

This question tests your understanding of the acceleration of a mass in SHM, specifically at the maximum displacement.

Key Terms and Formulas

• Spring constant: $k$

• Mass: $m$

• Acceleration in SHM: $a = -\frac{k}{m} x$

Step-by-Step Guidance

1. Recall the formula for acceleration in SHM: $a = -\frac{k}{m} x$

2. Identify the values: $k = 50$ N/m, $m = 2$ kg.

3. Note that acceleration depends on displacement $x$; maximum acceleration occurs at maximum $x$.

4. Set up the formula for maximum acceleration: $a_{max} = \frac{k}{m} x_{max}$

Try solving on your own before revealing the answer!

Final Answer: $a_{max} = 25 x_{max}$

Maximum acceleration is $25$ times the maximum displacement.

Q3. A mass m is suspended from a spring. What is the equilibrium position?

Background

Topic: Static Equilibrium and Hooke's Law

This question tests your understanding of how to find the equilibrium position of a mass attached to a spring, where forces are balanced.

Key Terms and Formulas

• Equilibrium: $F_{spring} = F_g$

• Hooke's Law: $F_{spring} = kx$

• Gravitational Force: $F_g = mg$

Step-by-Step Guidance

1. Set up the equilibrium equation: $k x = m g$

2. Rearrange to solve for $x$: $x = \frac{m g}{k}$

3. Interpret $x$ as the distance the spring stretches from its natural length.

Try solving on your own before revealing the answer!

Final Answer: $x = \frac{m g}{k}$

The equilibrium position is where the spring stretches by $\frac{m g}{k}$.

Q4. Someone swings a string in a horizontal circle. What is the acceleration?

Background

Topic: Circular Motion

This question tests your understanding of centripetal acceleration in circular motion.

Key Terms and Formulas

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

• Velocity: $v$

• Radius: $r$

Step-by-Step Guidance

1. Identify the formula for centripetal acceleration: $a_c = \frac{v^2}{r}$

2. Determine the values for $v$ and $r$ (if given).

3. Plug the values into the formula to set up the calculation.

Try solving on your own before revealing the answer!

Final Answer: $a_c = \frac{v^2}{r}$

Centripetal acceleration depends on the square of the velocity and the radius of the circle.

Q5. Two blocks are connected by a string over a pulley. What is the acceleration of the system?

Background

Topic: Newton's Second Law and Pulley Systems

This question tests your understanding of how to analyze forces in a pulley system and calculate the acceleration of connected masses.

Key Terms and Formulas

• Newton's Second Law: $F = m a$

• Force difference: $F_{net} = m_2 g - m_1 g$ (if $m_2 > m_1$)

• Total mass: $m_1 + m_2$

• Acceleration: $a = \frac{F_{net}}{m_{total}}$

Step-by-Step Guidance

1. Draw a free-body diagram for each block.

2. Write Newton's Second Law for each block.

3. Set up the equation for net force: $F_{net} = m_2 g - m_1 g$

4. Calculate the total mass: $m_{total} = m_1 + m_2$

Try solving on your own before revealing the answer!

Final Answer: $a = \frac{(m_2 - m_1) g}{m_1 + m_2}$

The acceleration depends on the difference in mass and the total mass.

Q6. A rotating disk has angular velocity $\omega$ and moment of inertia $I$. What is its rotational kinetic energy?

Background

Topic: Rotational Kinetic Energy

This question tests your understanding of how to calculate the kinetic energy of a rotating object using angular velocity and moment of inertia.

Key Terms and Formulas

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

• Moment of inertia: $I$

• Angular velocity: $\omega$

Step-by-Step Guidance

1. Recall the formula for rotational kinetic energy: $K = \frac{1}{2} I \omega^2$

2. Identify the values for $I$ and $\omega$.

3. Set up the calculation by plugging in the values.

Try solving on your own before revealing the answer!

Final Answer: $K = \frac{1}{2} I \omega^2$

Rotational kinetic energy depends on the moment of inertia and the square of the angular velocity.

Q7. A block slides down a frictionless incline. What is its speed at the bottom?

Background

Topic: Conservation of Energy

This question tests your understanding of energy conservation, specifically how potential energy converts to kinetic energy.

Key Terms and Formulas

• Potential energy: $U = m g h$

• Kinetic energy: $K = \frac{1}{2} m v^2$

• Conservation of energy: $U_{initial} = K_{final}$

Step-by-Step Guidance

1. Set up the conservation of energy equation: $m g h = \frac{1}{2} m v^2$

2. Cancel $m$ from both sides.

3. Rearrange to solve for $v$: $v = \sqrt{2 g h}$

Try solving on your own before revealing the answer!

Final Answer: $v = \sqrt{2 g h}$

The speed at the bottom depends on the height and gravity.