Physics Final Exam Study Guide: Motion, Forces, Energy, and Momentum

Unit Conversions

Converting Units in Physics

Unit conversions are essential in physics to ensure consistency and accuracy in calculations. Common conversions include grams to micrograms, meters to centimeters, and seconds to minutes.

• Key Point: To convert grams to micrograms, multiply by $10^6$.

• Example: $0.000114$ g/day $= 0.000114 \times 10^6 = 114$ micrograms/day.

Motion in One Dimension

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. It is a vector quantity and can be positive or negative depending on direction.

• Formula: $v_{avg} = \frac{\Delta x}{\Delta t}$

• Example: If a dog runs 8.2 km away and returns in 25 min, the displacement for the return is $-8.2$ km, and $\Delta t = 1500$ s. $v_{avg} = \frac{-8200}{1500} = -5.47$ m/s.

Velocity-Time Graphs & Acceleration

Velocity-time graphs are used to analyze motion and determine acceleration. The slope of the graph at any point gives the instantaneous acceleration.

• Formula: $a = \frac{\Delta v}{\Delta t}$

• Example: If velocity increases from 0 to 7 m/s in 25 ms, $a = \frac{7}{0.025} = 280$ m/s2.

Circular Motion

Uniform Circular Motion

Objects moving in a circle at constant speed experience centripetal acceleration directed toward the center of the circle.

• Speed Formula: $v = \frac{2\pi r}{T}$

• Centripetal Acceleration: $a = \frac{v^2}{r}$

• Example: For a satellite in geostationary orbit, use Earth's radius and orbital radius to calculate $v$ and $a$.

Period and Frequency in Circular Motion

The period is the time taken for one complete revolution, and frequency is the number of revolutions per unit time.

• Period Formula: $T = \frac{2\pi r}{v}$

• Frequency Formula: $f = \frac{1}{T}$

• Example: To generate 9g acceleration in a centrifuge, set $a = 9g$ and solve for $f$ or RPM.

Banked Curves

Banked curves allow vehicles to turn safely at higher speeds by providing a component of normal force toward the center of the curve.

• Formula: $v = \sqrt{r g \tan \theta}$ (without friction)

• With friction: $v_{max} = \sqrt{r g (\tan \theta + \mu_s)}$

• Example: For a curve of radius 85 m banked at 28°, calculate the maximum safe speed.

Forces & Newton's Laws

Free-Body Diagrams

Free-body diagrams are graphical representations of all the forces acting on an object. They are essential for analyzing equilibrium and motion.

• Key Forces: Tension (T), Weight (W), Normal force, Friction.

• Example: A soldier hanging from a rope experiences tension upward and weight downward. If accelerating horizontally, tension is angled.

Static Friction

Static friction prevents objects from sliding. The maximum static friction force is proportional to the normal force.

• Formula: $f_{s,max} = \mu_s N$

• Minimum Coefficient: $\mu_s = \frac{a}{g}$ (for horizontal acceleration)

• Example: For $a = 0.5g$, $\mu_s = 0.5$.

Rotational Motion

Types of Acceleration in Rotation

Rotational motion involves angular acceleration, which is the rate of change of angular velocity.

• Formula: $\alpha = \frac{\Delta \omega}{\Delta t}$

• Example: For a toy sled on a thread, calculate the number of revolutions before the thread snaps using torque and angular acceleration.

Converting Between Linear & Rotational Quantities

Linear and rotational motion are related through the radius of rotation.

• Formula: $v = \omega r$

• Length of Spiral Pattern: $L = v \times t$

• Example: For a printing head moving at 1.80 m/s for 42 min, $L = 1.80 \times 2520 = 4536$ m.

Equilibrium & Elasticity

Beams and Shelves Against a Wall

Analyzing equilibrium involves balancing forces and torques. Free-body diagrams help determine tension and reaction forces.

• Torque Formula: $\tau = r F \sin \theta$

• Example: For a rod with a decoration, set up equations for vertical and horizontal equilibrium and solve for tension and force direction.

Center of Mass

The center of mass is the point where the mass of a system is concentrated. For discrete masses, it is calculated using weighted averages.

• Formula: $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

• Example: For two marbles connected by a rod, calculate the distance from the 400 g marble to the center of mass.

Momentum

Impulse with Variable Forces

Impulse is the change in momentum, equal to the area under a force-time graph.

• Formula: $J = \int F(t) dt$

• Example: For a triangular force-time graph, calculate the area to find impulse.

Push-Away Problems (Conservation of Momentum)

In explosions or separations, total momentum before and after must be equal.

• Formula: $\sum m_i v_i = \sum m_f v_f$

• Example: For a rocket breaking into parts, sum the momenta of all parts to find initial velocity.

Adding Mass to a Moving System

When objects stick together after a collision, use conservation of momentum to find final velocity.

• Formula: $v_f = \frac{m_1 v_1 + m_2 v_2 + m_3 v_3}{m_1 + m_2 + m_3}$

• Example: For a bus, truck, and car colliding and sticking together, substitute masses and velocities.

Intro to Momentum (Vector Components)

Momentum is a vector quantity. The total momentum of a system is the vector sum of individual momenta.

• Formula: $\vec{p} = m \vec{v}$

• Example: For an SUV and truck moving in perpendicular directions, calculate $p_x$ and $p_y$ components.

Energy & Work

Energy in Connected Objects (Systems)

When objects are connected by pulleys and springs, use conservation of energy to analyze motion.

• Formula: $\frac{1}{2} k x^2 = m g h$

• Example: For a box compressing a spring, solve for $x$ (compression distance).

Conservation of Energy with Rotation

Rotational kinetic energy and gravitational potential energy are interconverted in rotating systems.

• Formula: $m g h = \frac{1}{2} I \omega^2$

• Example: For a bar rotating from vertical to horizontal, use moment of inertia for a rod about one end: $I = \frac{1}{3} m L^2$.

Power

Power is the rate at which work is done or energy is transferred.

• Formula: $P = F v$

• Example: For a cart pulled up an incline with friction, calculate total force and multiply by velocity.

Elastic Collisions

Conservation of Momentum and Energy

In elastic collisions, both momentum and kinetic energy are conserved.

• Momentum Conservation: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

• Kinetic Energy Conservation: $\frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$

• Example: For two cars colliding, solve the system of equations for final velocities.