Physics Study Guide: Mechanics and Applications

Mechanics: Core Concepts and Applications

Center of Mass

The center of mass of a system is the point at which the mass of the system can be considered to be concentrated for the purposes of analyzing translational motion. For composite objects, the center of mass is found by averaging the positions of all mass elements, weighted by their masses.

• Definition: The center of mass (COM) of a system of particles is given by: $\vec{R}_{\text{COM}} = \frac{1}{M} \sum_{i} m_i \vec{r}_i$ where $M$ is the total mass, $m_i$ is the mass of the $i$th particle, and $\vec{r}_i$ is its position vector.

• Example: For a uniform metal plate shaped as a cube with side length $L$, the center of mass is at the geometric center: $\left(\frac{L}{2}, \frac{L}{2}, \frac{L}{2}\right)$.

Space Vehicle Motion

When a space vehicle expels exhaust, it demonstrates the principle of conservation of momentum. The final speed of the vehicle can be determined using the rocket equation.

• Rocket Equation: $v_f = v_e \ln\left(\frac{m_i}{m_f}\right)$ where $v_f$ is the final speed, $v_e$ is the exhaust velocity, $m_i$ is the initial mass, and $m_f$ is the final mass.

• Application: Used to calculate the speed of a rocket after burning fuel and ejecting exhaust mass.

Energy in a Spring

Springs store energy when compressed or stretched, described by Hooke's Law. The energy stored is called elastic potential energy.

• Hooke's Law: $F = -kx$ where $F$ is the restoring force, $k$ is the spring constant, and $x$ is the displacement from equilibrium.

• Elastic Potential Energy: $U = \frac{1}{2}kx^2$ This is the energy stored in a spring when displaced by $x$.

• Example: Two blocks connected by a compressed spring will move apart when the spring is released, converting potential energy into kinetic energy.

Collisions: Conservation of Momentum

Collisions are analyzed using the conservation of momentum and, in some cases, conservation of energy. Collisions can be elastic or inelastic.

• Conservation of Momentum: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$ where $m_1, m_2$ are masses and $v_{1i}, v_{2i}, v_{1f}, v_{2f}$ are initial and final velocities.

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

• Inelastic Collision: Only momentum is conserved; kinetic energy is not.

• Example: A 5.0 kg block collides with a 10.0 kg block at rest. After the collision, the blocks move with new velocities determined by conservation laws.

Colliding with a Spring Object

When an object collides with another attached to a spring, both conservation of momentum and energy principles may apply, depending on friction and elasticity.

• Key Principle: If the collision is perfectly inelastic, the objects stick together and move as one mass after the collision.

• Spring Compression: The maximum compression of the spring can be found by equating the initial kinetic energy to the elastic potential energy at maximum compression: $\frac{1}{2} m v^2 = \frac{1}{2} k x^2$ where $x$ is the maximum compression.

Vector Notation in Mechanics

Vectors are used to represent quantities with both magnitude and direction, such as velocity and force. Vector notation simplifies calculations in two or three dimensions.

• Vector Addition: $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$

• Example: If two blocks collide, their velocities before and after the collision can be expressed as vectors, and the final velocity is found by applying conservation of momentum in vector form.

Jumping the Gap: Friction and Motion

When a car jumps a gap between two surfaces, the analysis involves friction, energy, and projectile motion. The width of the gap can be determined by considering the car's speed and the forces acting on it.

• Frictional Force: $F_f = \mu N$ where $\mu$ is the coefficient of friction and $N$ is the normal force.

• Projectile Motion: The horizontal distance (range) is given by: $R = \frac{v^2 \sin 2\theta}{g}$ where $v$ is the speed, $\theta$ is the launch angle, and $g$ is acceleration due to gravity.

• Application: Used to determine if the car can successfully jump the gap.

Hot Air Balloon: Forces and Relative Motion

Hot air balloons demonstrate principles of buoyancy and relative motion. When a person climbs a ladder attached to a balloon, the system's center of mass and velocity change.

• Buoyant Force: $F_b = \rho_{\text{air}} V g$ where $\rho_{\text{air}}$ is air density, $V$ is volume, and $g$ is gravity.

• Relative Motion: If a person climbs up, the balloon moves in the opposite direction to conserve momentum.

• Example: A 78 kg man climbs a ladder attached to a 300 kg balloon. The balloon moves downward as the man climbs upward.

Summary Table: Collision Types

The following table summarizes the differences between elastic and inelastic collisions:

Collision Type	Momentum Conserved?	Kinetic Energy Conserved?	Example
Elastic	Yes	Yes	Billiard balls colliding
Inelastic	Yes	No	Clay balls sticking together