Midterm practice 2

Linear and Angular Momentum

Linear Momentum

Linear momentum is a measure of an object's motion, defined as the product of its mass and velocity. It is a vector quantity, meaning it has both magnitude and direction.

• Definition: $p = m v$ where $p$ is momentum, $m$ is mass, and $v$ is velocity.

• Units: kg·m/s

• Conservation: In the absence of external forces, the total linear momentum of a system remains constant.

• Example: A 10 kg object moving at 5 m/s has a momentum of $p = 10 \times 5 = 50$ kg·m/s.

Angular Momentum

Angular momentum is the rotational analog of linear momentum, associated with objects rotating about an axis.

• Definition: $L = I \omega$ where $L$ is angular momentum, $I$ is moment of inertia, and $\omega$ is angular velocity.

• Units: kg·m2/s

• Conservation: Angular momentum is conserved in a system with no external torque.

• Example: A spinning disk with $I = 2$ kg·m2 and $\omega = 3$ rad/s has $L = 6$ kg·m2/s.

Torque and Rotational Dynamics

Torque

Torque is a measure of the tendency of a force to rotate an object about an axis.

• Definition: $\tau = r F \sin \theta$ where $r$ is the lever arm, $F$ is the force, and $\theta$ is the angle between force and lever arm.

• Direction: Counterclockwise torques are considered positive; clockwise are negative.

• Example: Applying a force at the end of a door creates a larger torque than at the middle.

Ordering Torques

When multiple forces act at different points and angles, compare their torques using the formula above. The greater the lever arm and the component of force perpendicular to the lever arm, the greater the torque.

Circular Motion and Centripetal Force

Centrifuge Acceleration

When a centrifuge spins up to a certain angular speed, its angular acceleration can be found using rotational kinematics.

• Angular acceleration: $\alpha = \frac{\omega_f - \omega_i}{t}$

• Number of revolutions: $\theta = \omega_i t + \frac{1}{2} \alpha t^2$ (in radians), then divide by $2\pi$ for revolutions.

• Example: If $\omega_f = 1500$ rad/s, $t = 5$ s, $\omega_i = 0$, then $\alpha = 300$ rad/s2.

Centripetal Force and Friction

For an object moving in a circle, the required centripetal force is provided by friction.

• Formula: $F_c = \frac{m v^2}{r}$

• Minimum coefficient of static friction: $\mu_s = \frac{v^2}{r g}$

• Example: A bug on a record player must have enough friction to avoid slipping.

Work, Energy, and Drag Forces

Work Done by Drag

When an object falls through the air, drag force does negative work, reducing its mechanical energy.

• Work-energy principle: $W_{\text{drag}} = \Delta KE + \Delta PE$

• Example: If a person falls from 500 m and lands at 10 m/s, calculate the change in kinetic and potential energy to find work done by drag.

Terminal Velocity and Drag

Terminal velocity is reached when the upward drag force equals the downward gravitational force.

• At terminal velocity: $F_{\text{drag}} = mg$

• Magnitude of drag force: $F_{\text{drag}} = \frac{1}{2} C_d \rho A v^2$

• Example: Opening a parachute increases $A$, decreasing terminal velocity and drag force.

Forces and Free Body Diagrams

Free Body Diagrams

Free body diagrams are essential for analyzing forces acting on an object.

• Identify all forces: Gravity, normal force, friction, tension, drag, etc.

• Application: For Harold being pulled by a kite, include gravity, normal force, friction, and tension from the kite.

• For the kite: Include tension, drag force from wind, and ignore mass if specified.

Statics and Equilibrium

Normal Forces and Supports

When a board is supported at two points and loaded with masses, use equilibrium conditions to solve for support forces.

• Sum of forces: $N_1 + N_2 = (M + m)g$

• Sum of torques about one support: $N_2 \times d_2 = M g \times d_M + m g \times d_m$

• Example: A bear sitting on a board changes the normal force at each support depending on its position.

Roller Coaster and Energy Conservation

Conservation of Energy in Roller Coasters

Roller coasters convert potential energy to kinetic energy and vice versa as they move along the track.

• Potential energy: $PE = mgh$

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

• Application: The coaster can reach any point lower than its starting height, neglecting friction.

Summary Table: Key Formulas and Concepts

Concept	Formula	Units
Linear Momentum	$p = m v$	kg·m/s
Angular Momentum	$L = I \omega$	kg·m2/s
Torque	$\tau = r F \sin \theta$	N·m
Centripetal Force	$F_c = \frac{m v^2}{r}$	N
Work-Energy Principle	$W = \Delta KE + \Delta PE$	J
Drag Force	$F_{\text{drag}} = \frac{1}{2} C_d \rho A v^2$	N
Normal Force (Statics)	$N_1 + N_2 = (M + m)g$	N

Additional info: These notes cover topics from motion, forces, energy, rotational dynamics, and statics, as relevant to a college-level physics course. The original questions have been expanded with definitions, formulas, and examples for comprehensive review.