General Physics I – Practice Midterm Exam 2 Study Guide

Problem A: Multiple Choice & Short Answer – Mechanics and Rotational Motion

Conservation of Energy and Mechanical Systems

This section covers the principles of energy conservation, work, and mechanical energy in physical systems, including blocks on slopes and frictional forces.

• Mechanical Energy Conservation: In the absence of non-conservative forces (like friction), the total mechanical energy (kinetic + potential) of a system remains constant.

• Work Done by Friction: Friction is a non-conservative force that dissipates mechanical energy as heat, reducing the system's total mechanical energy.

• Key Equation: $E_{mech, initial} = E_{mech, final} + E_{dissipated}$

• Example: A block sliding down a slope loses energy to friction, so its final kinetic energy is less than the initial potential energy.

Power and Work in Frictional Systems

Power is the rate at which work is done, especially relevant when pushing objects against friction.

• Power Formula: $P = F_{friction} \cdot v$

• Frictional Force: $F_{friction} = \mu mg$

• Example: Pushing a box at constant velocity requires a force equal to friction, and the power is the product of this force and velocity.

Rotational Motion and Angular Velocity

Rotational motion involves objects moving in circles, characterized by angular velocity and moment of inertia.

• Angular Velocity ($\omega$): The rate of change of angular displacement, measured in radians per second.

• Formula for Turntable: $\omega = \frac{mvr}{I}$

• Example: A bottle slides on a turntable, and its motion affects the turntable's angular velocity due to conservation of angular momentum.

Circular Motion and Revolution Calculations

Calculating the number of revolutions made by a wheel or object moving in a circle.

• Number of Revolutions: $N = \frac{\text{Distance}}{2\pi r}$

• Example: A cyclist rides 5 miles; the number of wheel revolutions depends on the wheel radius.

Rolling Motion and Work

Work required to stop a rolling object involves both translational and rotational kinetic energy.

• Kinetic Energy of Rolling Cylinder: $KE_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

• Work to Stop: Equal to the total initial kinetic energy.

• Moment of Inertia for Solid Cylinder: $I = \frac{1}{2}mr^2$

Gravitational Orbits and Satellite Motion

Calculating the altitude for a satellite in geosynchronous orbit using Newton's law of gravitation and orbital mechanics.

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

• Orbital Period: $T = 2\pi \sqrt{\frac{r^3}{GM}}$

• Application: Determining the required altitude for a satellite to match a planet's rotation period.

Mechanical Energy, Impulse, and Scattering

Identifying mechanical energy, linear momentum, and impulse in various physical scenarios.

• Impulse: Change in momentum due to a force applied over time.

• Scattering: Neutron scattering off a proton involves conservation of momentum and energy.

• Example: A ball falling and bouncing, a skater pulling arms in to spin faster (conservation of angular momentum).

Problem B: Forces and Equilibrium – Tension in Wires

Free Body Diagrams (FBD) and Tension Analysis

Analyzing forces acting on a suspended bucket using free body diagrams and equilibrium conditions.

• Free Body Diagram: Shows all forces acting on the bucket, including tensions in wires and gravity.

• Equilibrium Conditions: The sum of forces in both x and y directions must be zero.

• Tension Calculation: Use trigonometry to resolve tensions in each wire: $T_1 \sin(\theta_1) + T_2 \sin(\theta_2) = mg$

• Application: Determining which wire breaks when maximum tension is reached, and calculating the mass of sand at breaking point.

Problem C: Springs and Energy – Hooke's Law and Energy Dissipation

Spring Compression and Hooke's Law

Understanding how springs compress under load and the energy stored in them.

• Hooke's Law: $F = kx$

• Spring Energy: $E = \frac{1}{2}kx^2$

• Equilibrium Compression: When a mass is placed, springs compress until the upward force equals the weight.

Energy Dissipation in Springs

Calculating energy lost to non-conservative forces when compression is less than expected.

• Energy Lost: Difference between initial and final elastic potential energy.

• Application: If compression is only 90% of expected, calculate lost energy.

Problem D: Rotational Motion – Pinball Flipper Mechanics

Rotational Inertia and Torque

Analyzing the rotational motion of a pinball flipper, including torque, angular acceleration, and energy transfer.

• Moment of Inertia ($I$): Resistance to rotational acceleration, depends on mass distribution.

• Torque ($\tau$): $\tau = I\alpha$

• Angular Acceleration ($\alpha$): Rate of change of angular velocity.

• Application: Calculating torque, angular acceleration, and energy transfer during pinball collisions.

Energy and Momentum in Collisions

Understanding how energy and angular momentum are transferred during collisions between the flipper and the pinball.

• Conservation of Angular Momentum: Total angular momentum before and after collision remains constant if no external torque acts.

• Kinetic Energy Change: Difference in kinetic energy before and after collision.

Key Formulas and Concepts Table

Concept	Formula	Application
Mechanical Energy	$E_{mech} = KE + PE$	Conservation in closed systems
Work by Friction	$W = F_{friction} \cdot d$	Energy lost to heat
Power	$P = F \cdot v$	Rate of work done
Angular Velocity	$\omega = \frac{v}{r}$	Rotational motion
Hooke's Law	$F = kx$	Spring compression
Spring Energy	$E = \frac{1}{2}kx^2$	Elastic potential energy
Torque	$\tau = rF \sin\theta$	Rotational force
Moment of Inertia	$I = \sum m_ir_i^2$	Rotational inertia
Angular Momentum	$L = I\omega$	Conservation in collisions

Additional info: These study notes expand upon the exam questions by providing definitions, formulas, and academic context for each topic, ensuring a comprehensive review for students preparing for a General Physics I midterm exam.