Physics 211: Midterm 3 Review – Momentum, Energy, Rotation, and Gravity

Momentum and Energy

Conservation Laws in Collisions

In physics, conservation laws are fundamental principles that govern the behavior of physical systems during interactions such as collisions. Momentum is always conserved in isolated systems, while mechanical energy may not be conserved in inelastic collisions.

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not. Some energy is transformed into other forms (e.g., heat, sound).

• Example: A car of mass m rolls down a hill from height h, collides inelastically with a stationary car of mass 2m, and the two stick together and move up another hill. Only a fraction of the initial kinetic energy remains after the collision.

Key Equations:

• Conservation of momentum: $m v = (m + 2m) v' = 3m v'$

• Final velocity after collision: $v' = \frac{v}{3}$

• Kinetic energy after collision: $K_f = \frac{1}{2} (m + 2m) v'^2 = \frac{1}{6} m v^2$

• Fraction of kinetic energy remaining: $\frac{K_f}{K_i} = \frac{1}{3}$

Energy and Power

Work, Energy, and Power in Vehicles

Power in vehicles is the rate at which energy is converted from fuel to mechanical work. The maximum speed a vehicle can achieve is limited by the available engine power, especially when climbing an incline.

• Power: $P = F v$

• Maximum speed up a hill: $v_{max} = \frac{P_{max}}{m g \sin \theta}$

• Example: For a 2,000 kg car with 100,000 W engine power on a $20^\circ$ incline: $v_{max} = \frac{100,000}{2,000 \times 9.8 \times \sin 20^\circ} = 15$ m/s

Rotational Kinematics

Equations for Constant Acceleration

Rotational kinematics describes the motion of rotating objects using angular analogs of linear motion equations.

• Linear (Translational) Motion:

  • $v = a t + v_0$

  • $x = \frac{1}{2} a t^2 + v_0 t + x_0$

  • $v^2 = v_0^2 + 2a(x - x_0)$

• Rotational (Angular) Motion:

  • $\omega = \alpha t + \omega_0$

  • $\theta = \frac{1}{2} \alpha t^2 + \omega_0 t + \theta_0$

  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Circular Motion

Centripetal Force and Turning Limits

Objects moving in a circle require a centripetal force directed toward the center of the circle. For cars, this force is provided by static friction.

• Centripetal force: $F_c = m \frac{v^2}{r}$

• Maximum speed in a turn: $v_{max} = \sqrt{\mu_s r g}$

• Static friction: $F_s = \mu_s N$

• Note: The maximum speed does not depend on the car's mass.

Gravity

Gravitational Force and Orbits

Gravity is the attractive force between masses, described by Newton's law of universal gravitation. It governs planetary motion and satellite orbits.

• Newton's Law of Gravitation: $F = G \frac{M m}{r^2}$

• Gravitational acceleration at Earth's surface: $g = G \frac{M}{r^2}$

• Orbital speed: $v = \sqrt{\frac{G M}{r}}$

• Escape velocity: $v_{esc} = \sqrt{\frac{2 G M}{r}}$

Torque

Rotational Equilibrium and Forces

Torque is the rotational analog of force, causing objects to rotate about an axis. For static equilibrium, both net force and net torque must be zero.

• Torque: $\tau = r F \sin \theta$

• Equilibrium conditions: $\sum F = 0$ and $\sum \tau = 0$

• Example: Ladder leaning against a wall: analyze forces and torques to find the minimum angle for equilibrium.

Moment of Inertia

Rotational Inertia and Parallel-Axis Theorem

The moment of inertia quantifies an object's resistance to changes in rotational motion. The parallel-axis theorem allows calculation of the moment of inertia about any axis parallel to one through the center of mass.

• Moment of inertia (rod about center): $I_{cm} = \frac{1}{12} m L^2$

• Parallel-axis theorem: $I = I_{cm} + m d^2$

• Example: Tall trees (long rods) fall slower due to larger moment of inertia.

Rotational Energy and Rolling

Kinetic Energy in Rolling Motion

When an object rolls without slipping, its total kinetic energy is the sum of translational and rotational energies.

• Total kinetic energy: $K_{total} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$

• No-slip condition: $v = r \omega$

• For a solid sphere: $I = \frac{2}{5} m r^2$

• For a hollow cylinder: $I = m r^2$

• Energy conservation down a hill: $mgh = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$

Applications and Examples

Summary Table: Key Equations and Concepts

Concept	Equation	Notes
Momentum Conservation	$m_1 v_1 + m_2 v_2 = (m_1 + m_2) v'$	Inelastic collision
Kinetic Energy (Rolling)	$K_{total} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$	Rolling without slipping
Power (Vehicle)	$P = F v$	Engine power limits speed
Gravitational Force	$F = G \frac{M m}{r^2}$	Newton's law
Torque	$\tau = r F \sin \theta$	Rotational equilibrium
Moment of Inertia (Rod)	$I_{cm} = \frac{1}{12} m L^2$	About center

Additional info: These notes expand on the provided slides and images, filling in academic context and definitions for clarity and completeness. All equations are presented in LaTeX format for clarity.