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

Mechanics: Motion and Forces

Motion on an Inclined Plane

Objects sliding down an inclined plane experience acceleration due to gravity, modified by the angle of the incline. The absence of friction simplifies calculations.

• Acceleration on an Incline: The acceleration $a$ of an object down a frictionless incline of angle $\theta$ is given by:

$a = g \sin\theta$

• Final Velocity: If an object starts from rest and slides a distance $d$ down the incline, its final velocity $v$ is:

$v = \sqrt{2ad}$

• Time to Slide Down: The time $t$ to slide distance $d$ is:

$t = \sqrt{\frac{2d}{a}}$

• Example: For a 5-meter ramp at 20°, $a = 9.8 \sin 20° \approx 3.35$ m/s2.

Friction and Kinetic Friction

Friction opposes motion between surfaces. The coefficient of kinetic friction $\mu_k$ quantifies this effect.

• Kinetic Friction Force:

$f_k = \mu_k N$

• Where $N$ is the normal force.

• Acceleration with Friction: For an object sliding down an incline with friction:

$a = g(\sin\theta - \mu_k \cos\theta)$

Projectile and Relative Motion

Relative motion problems involve vector addition and trigonometry to determine net displacement.

• Example: Walking in multiple directions requires breaking each segment into components and summing them.

Rotational Motion and Energy

Moment of Inertia

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

• 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: For a solid sphere of radius 0.5 m, mass $M$, and $\omega = 80$ rad/s:

$K_{rot} = \frac{1}{2} \left(\frac{2}{5}MR^2\right)\omega^2$

Angular Momentum

Angular momentum $L$ is a measure of rotational motion, conserved in isolated systems.

• Formula:

$L = I\omega$

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

Rolling Motion and Energy

When objects roll without slipping, both translational and rotational kinetic energies are present.

• Total Kinetic Energy:

$K_{total} = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2$

• Objects Rolling Down an Incline: Objects with lower moments of inertia reach the bottom faster.

• Example: A solid sphere rolls faster than a hollow sphere of the same mass and radius.

Work, Energy, and Springs

Work and Energy Conservation

Work done on an object changes its kinetic energy. For conservative forces, mechanical energy is conserved.

• Work-Energy Theorem:

$W_{net} = \Delta K$

• Potential Energy in Springs (Hooke's Law):

$U = \frac{1}{2}kx^2$

• Where $k$ is the spring constant and $x$ is the compression or extension.

• Example: To find $x$ for a given $U$ and $k$:

$x = \sqrt{\frac{2U}{k}}$

Collisions and Conservation Laws

Conservation of Momentum

In the absence of external forces, the total momentum of a system remains constant.

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

• Inelastic Collisions: Only momentum is conserved.

• Example: Two rocks of equal mass collide and bounce off; use conservation laws to find final 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.

• Acceleration in Pulley Systems: For a mass $m$ hanging from a frictionless pulley:

$a = \frac{mg}{m + M}$

• Where $M$ is the mass of the other object or system.

• Example: Two weights attached to a pulley; calculate acceleration using force balance.

Terminal Velocity and Drag

Terminal Velocity

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

• Factors Affecting Terminal Velocity: Mass, cross-sectional area, and drag coefficient.

• Example: Larger, heavier spheres generally have higher terminal velocities if drag is proportional to velocity.

Open-Ended Problem-Solving

Vector Addition and Resultant Forces

When multiple forces act on an object, the net force is found by vector addition. The acceleration is then:

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

Space and Projectile Problems

Throwing objects in space demonstrates conservation of momentum. The direction and speed of the throw determine the resulting motion.

Rotational Dynamics and Torque

Applying brakes to a wheel involves torque and friction. The net torque is related to the friction force at the ground:

$\tau = rF_{friction}$

• If the mass of the wheel is zero, the moment of inertia is zero, and the net torque must also be zero.

Summary Table: Key Rotational Quantities

Quantity	Symbol	Formula	Units
Moment of Inertia (solid sphere)	I	$\frac{2}{5}MR^2$	kg·m2
Rotational Kinetic Energy	$K_{rot}$	$\frac{1}{2}I\omega^2$	Joules (J)
Angular Momentum	L	$I\omega$	kg·m2/s
Spring Potential Energy	U	$\frac{1}{2}kx^2$	Joules (J)

Additional info: These notes synthesize key concepts from the provided practice questions, expanding on the underlying physics principles and equations relevant to introductory college mechanics and rotational motion.