Physics 1403 Practice Final: Study Notes on Motion, Forces, Energy, Rotation, and Friction

Motion and Kinematics

Motion on an Inclined Plane

Objects sliding down a frictionless ramp experience constant acceleration due to gravity, modified by the ramp's angle.

• Acceleration down a ramp: $a = g \sin\theta$, where $g$ is the acceleration due to gravity and $\theta$ is the ramp angle.

• Final velocity: For an object starting from rest and traveling a distance $d$ down the ramp, $v = \sqrt{2ad}$.

• Time to travel down the ramp: $t = \sqrt{\frac{2d}{a}}$.

• Example: For a 20-degree ramp, $a = 9.8 \sin(20^\circ) \approx 3.35\,\text{m/s}^2$.

Displacement and Relative Position

Displacement is a vector quantity representing the shortest path between initial and final positions.

• Vector addition: Use the Pythagorean theorem and trigonometry to find resultant displacement.

• Example: Walking north, east, and then at an angle requires breaking each segment into components and summing them.

Forces and Newton's Laws

Newton's Second Law

Newton's Second Law relates force, mass, and acceleration.

• Formula: $\vec{F} = m\vec{a}$

• Application: Used to calculate acceleration when multiple forces act on an object.

Friction

Friction opposes motion and is characterized by coefficients of static and kinetic friction.

• Kinetic friction force: $f_k = \mu_k N$, where $\mu_k$ is the coefficient of kinetic friction and $N$ is the normal force.

• Static friction force: $f_s \leq \mu_s N$

• Example: Car tires on a track, where the coefficient of friction determines the maximum possible acceleration.

Rotational Motion and Energy

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in rotational motion.

• Formula for a solid sphere: $I = \frac{2}{5}MR^2$, where $M$ is mass and $R$ is radius.

• Rotational kinetic energy: $K_{rot} = \frac{1}{2}I\omega^2$, where $\omega$ is angular velocity.

• Example: Calculating rotational energy for a sphere spinning at a given rate.

Angular Momentum

Angular momentum is a measure of rotational motion, dependent on moment of inertia and angular velocity.

• Formula: $L = I\omega$

• Conservation: In the absence of external torques, angular momentum is conserved.

Energy in Springs

Springs store mechanical energy when compressed or stretched.

• Spring potential energy: $U = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement.

• Solving for compression: $x = \sqrt{\frac{2U}{k}}$

Collisions and Conservation Laws

Conservation of Momentum

In collisions, the total momentum of a system is conserved if no external forces act.

• Formula: $m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$

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

• Inelastic collisions: Only momentum is conserved.

Terminal Velocity and Drag

Terminal Velocity

Terminal velocity is the constant speed reached by an object when the force of gravity is balanced by drag.

• Formula: $v_t = \sqrt{\frac{2mg}{\rho AC_d}}$, where $m$ is mass, $g$ is gravity, $\rho$ is fluid density, $A$ is cross-sectional area, and $C_d$ is drag coefficient.

• Factors affecting terminal velocity: Mass, shape, and area of the object.

Work, Energy, and Power

Work and Energy

Work is done when a force moves an object over a distance.

• Formula: $W = Fd\cos\theta$

• Kinetic energy: $K = \frac{1}{2}mv^2$

• Potential energy: $U = mgh$

Sample Problems and Applications

Sample Calculations

• Acceleration vector: For multiple forces, use vector addition to find net force and divide by mass.

• Projectile motion: Use kinematic equations to determine range, time, and final velocity.

• Pulley systems: Apply Newton's laws, considering tension and mass distribution.

Tables

Comparison of Rotational Quantities

Quantity	Formula	Physical Meaning
Moment of Inertia (Solid Sphere)	$I = \frac{2}{5}MR^2$	Resistance to rotational acceleration
Rotational Kinetic Energy	$K_{rot} = \frac{1}{2}I\omega^2$	Energy due to rotation
Angular Momentum	$L = I\omega$	Rotational analog of linear momentum

Terminal Velocity Factors

Object	Mass	Diameter	Terminal Velocity
Sphere A	1000 kg	1 m	Higher (due to greater mass)
Sphere B	1000 kg	4 m	Lower (due to larger area)
Sphere C	2000 kg	1 m	Highest (due to greatest mass)
Sphere D	2000 kg	4 m	Lowest (due to largest area)

Additional info:

• Some questions involve vector addition and trigonometry for displacement calculations.

• Open-ended questions require application of Newton's laws, kinematics, and rotational dynamics.

• Students should be familiar with basic equations of motion, energy, and rotational quantities.