Physics 1403 Practice Final: Study Notes on Mechanics, Energy, and Rotational Motion

Mechanics: Motion and Forces

Motion on an Inclined Plane

When an object slides down a frictionless ramp, its acceleration and velocity can be determined using Newton's laws and kinematics.

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

• Final velocity after traveling distance $d$: $v = \sqrt{2ad}$

• Time to travel down the ramp: $d = \frac{1}{2} a t^2 \implies t = \sqrt{\frac{2d}{a}}$

• Example: For a 20-degree incline and a 5-meter ramp, calculate $a$ and then $v$ and $t$.

Friction and Kinetic Friction

Friction opposes the motion of objects sliding on surfaces. The coefficient of kinetic friction ($\mu_k$) determines the frictional force.

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

• Acceleration with friction: $a = g(\sin\theta - \mu_k \cos\theta)$

• Example: For a ramp with $\mu_k = 0.3$, calculate the acceleration of a sliding object.

Projectile and Relative Motion

Relative motion problems involve determining displacement using vector addition.

• Displacement: Use the Pythagorean theorem and vector components to find the resultant displacement.

• Example: Walking north, east, and then at an angle; sum the vectors to find the net displacement from the starting point.

Rotational Motion and Energy

Moment of Inertia

The moment of inertia ($I$) quantifies an object's resistance to rotational acceleration about an axis.

• Solid sphere: $I = \frac{2}{5} m r^2$

• Rotational kinetic energy: $K_{rot} = \frac{1}{2} I \omega^2$

• Example: For a sphere of mass $m$ and radius $r$ spinning at angular velocity $\omega$, calculate $K_{rot}$.

Angular Momentum

Angular momentum ($L$) is conserved in the absence of external torques.

• Formula: $L = I \omega$

• Example: Comparing spheres of different radii, masses, and angular velocities to determine which has the greatest angular momentum.

Energy in Springs

Springs store potential energy when compressed or stretched.

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

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

• Example: For $U = 2250$ J and $k = 500$ N/m, find $x$.

Collisions and Conservation Laws

Conservation of Momentum

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

• Momentum: $p = mv$

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

• Example: Two rocks of equal mass collide; use conservation laws to find post-collision velocities.

Equilibrium and Forces

Pulleys and Tension

Pulley systems are used to change the direction of forces and can be analyzed using Newton's laws.

• Forces on masses: $T - mg = ma$ for hanging masses, where $T$ is tension.

• Massless, frictionless pulley: The tension is the same on both sides of the rope.

• Example: Two weights attached to a pulley; solve for acceleration using force equations.

Terminal Velocity and Drag

Terminal Velocity

Terminal velocity is reached when the drag force equals the gravitational force, resulting in zero net acceleration.

• Drag force: $F_{drag} = \frac{1}{2} C_d \rho A v^2$

• Terminal velocity: $v_t = \sqrt{\frac{2mg}{C_d \rho A}}$

• Example: Comparing spheres of different sizes and masses to determine which has the largest terminal velocity.

Open-Ended Problem Solving

Force and Acceleration Vectors

When multiple forces act on an object, the net force and resulting acceleration can be found using vector addition.

• Net force: $\vec{F}_{net} = \vec{F}_1 + \vec{F}_2 + ...$

• Acceleration: $\vec{a} = \frac{\vec{F}_{net}}{m}$

Space and Projectile Motion

Throwing an object in space demonstrates conservation of momentum; the astronaut moves in the opposite direction to the thrown object.

• Conservation of momentum: $m_1 v_1 = -m_2 v_2$

Rotational Dynamics and Torque

Applying a force at a distance from the axis of rotation creates torque, which changes the rotational motion of an object.

• Torque: $\tau = r F$

• Relation to friction: The friction force at the ground creates a torque that slows the wheel when braking a bicycle.

Summary Table: Key Rotational Quantities

Quantity	Symbol	Formula	Units
Moment of Inertia (solid sphere)	I	$\frac{2}{5} m r^2$	kg·m2
Rotational Kinetic Energy	$K_{rot}$	$\frac{1}{2} I \omega^2$	Joules (J)
Angular Momentum	L	$I \omega$	kg·m2/s
Torque	$\tau$	$r F$	Newton·meter (N·m)

Additional info: Some explanations and formulas have been expanded for clarity and completeness, including the explicit forms of equations for inclined planes, friction, and rotational motion.