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

Momentum and Energy

Conservation Principles

Understanding what physical quantities are conserved in different scenarios is fundamental in physics. Conservation laws help predict the outcomes of collisions and motion.

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

• Momentum Conservation: In isolated systems, total momentum is conserved, especially during collisions.

• Example: A car rolling down a hill (mechanical energy conserved), colliding inelastically with another car (momentum conserved, but not mechanical energy), and then moving uphill (mechanical energy conserved for the combined mass).

Energy and Power

Power in Vehicles

Power is the rate at which energy is converted or transferred. In vehicles, engines convert chemical energy (from fuel) into kinetic energy.

• Definition: Power ($P$) is the rate of doing work: $P = \frac{dW}{dt}$.

• Maximum Power: The maximum speed up a hill is limited by the available engine power.

• Formula: $P_{\text{max}} = F_{\text{min}} v_{\text{max}} = mg \sin \theta \cdot v_{\text{max}}$

• Example: For a 2,000 kg car with 100 kW engine on a $20^\circ$ incline: $v_{\text{max}} = \frac{P_{\text{max}}}{mg \sin \theta} = 15\ \text{m/s}$

Rotational Kinematics

1D Kinematics for Constant Acceleration

Kinematic equations describe motion under constant acceleration for both linear and rotational cases.

• Linear Motion:

  • $v = at + v_0$

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

  • $x = \frac{1}{2}(v + v_0)t + x_0$

  • $v^2 = v_0^2 + 2a(x - x_0)$

• Rotational Motion:

  • $\omega = \alpha t + \omega_0$

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

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

• Correspondence: $x \leftrightarrow s$, $v \leftrightarrow r\omega$, $a \leftrightarrow r\alpha$

Circular Motion

Turning and Centripetal Force

When a car turns, friction provides the centripetal force required for circular motion.

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

• Maximum Static Friction: $f_s = \mu_s N = \mu_s mg$

• Maximum Speed: $v_{\text{max}} = \sqrt{\mu_s g r}$

• Note: The result does not depend on the mass of the car.

Gravity

Gravitational Acceleration and Potential Energy

Gravity governs the motion of objects near Earth and in orbit. The gravitational acceleration $g$ can be calculated using Newton's law of universal gravitation.

• Gravitational Force: $F = G \frac{Mm}{r^2}$

• Gravitational Acceleration: $g = G \frac{M}{r^2}$

• On Earth: $g = 9.82\ \text{m/s}^2$

• Gravitational Potential Energy: $\Delta U = mgh$ (near Earth's surface)

• General Case: $W = -G M m \left( \frac{1}{r_f} - \frac{1}{r_i} \right)$

Rotational Dynamics

Torque and Equilibrium

Torque is the rotational equivalent of force and is crucial for analyzing equilibrium and rotational motion.

• Torque: $\tau = r F \sin \phi$

• Static Equilibrium: $\vec{F}_{\text{net}} = 0$ and $\vec{\tau}_{\text{net}} = 0$

• Example: Ladder leaning against a wall, airplane in level flight.

Moment of Inertia

Parallel-Axis Theorem

The moment of inertia quantifies an object's resistance to rotational acceleration. The parallel-axis theorem allows calculation of the moment of inertia about any axis.

• Parallel-Axis Theorem: $I = I_{\text{cm}} + Md^2$

• Application: Used for rods, disks, and composite bodies.

Rotational Energy and Rolling

Rolling Without Slipping

When an object rolls without slipping, both translational and rotational kinetic energies must be considered.

• No-Slip Condition: $v = r\omega$

• Kinetic Energy: $K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

• Ratio of Energies: $\frac{K_r}{K_t} = \beta$ where $\beta = \frac{I}{mr^2}$

• Rolling Down a Hill: $mgh = \frac{1+\beta}{2}mv^2$ so $v = \sqrt{\frac{2gh}{1+\beta}}$

Applications and Examples

Minimum Speed for Vertical Circular Motion

To complete a vertical loop, an object must have sufficient speed at the top to maintain contact.

• Condition: $F_c^{\text{top}} = m \frac{v^2}{r} = N_t + mg$

• Minimum Speed: $v_{\text{min}} = \sqrt{g r}$

Rolling Around a Loop

For a rolling object to complete a loop, both translational and rotational energies are considered.

• Required Height: $h' = h - 2R$

• For a marble ($\beta = \frac{2}{5}$): $h = \frac{27}{10}R$

• For a hollow cylinder ($\beta = 1$): $h = 3R$

Summary Table: Key Equations and Concepts

Concept	Equation	Notes
Mechanical Energy	$E = K + U$	Conserved if no non-conservative forces
Momentum	$p = mv$	Conserved in collisions
Power	$P = Fv$	Rate of energy transfer
Torque	$\tau = rF\sin\phi$	Rotational effect of force
Moment of Inertia	$I = I_{\text{cm}} + Md^2$	Parallel-axis theorem
Gravitational Force	$F = G\frac{Mm}{r^2}$	Newton's law of gravity
Escape Velocity	$v_{\text{escape}} = \sqrt{\frac{2GM}{r_E}}$	Minimum speed to leave Earth's gravity
Rolling Kinetic Energy	$K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$	For rolling without slipping

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