Chapter 7: Rotational Motion

Angular Velocity and Rotational Motion Graphs

Rotational motion describes the movement of objects around a fixed axis. Angular velocity quantifies how quickly an object rotates.

• Angular velocity (\(\omega\)): The rate of change of angular displacement, measured in radians per second (rad/s).

• Interpreting motion graphs: Graphs of angular displacement, velocity, and acceleration versus time reveal information about rotational motion, similar to linear motion graphs.

• Formula:

• Example: A wheel rotates 10 radians in 2 seconds. Its average angular velocity is rad/s.

Rotational Kinematic Equations

These equations describe rotational motion with constant angular acceleration, analogous to linear kinematics.

• Where \(\omega\) is angular velocity, \(\alpha\) is angular acceleration, and \(\theta\) is angular displacement.

• Example: If a disk starts from rest and accelerates at 2 rad/s2 for 3 s, rad/s.

Linear and Rotational Variables Comparison

Rotational variables have direct analogs in linear motion.

Period and Frequency

The period is the time for one complete rotation; frequency is the number of rotations per second.

• Example: If a wheel completes 5 rotations per second, s.

Angular and Tangential Acceleration

Angular acceleration (\(\alpha\)) is the rate of change of angular velocity. Tangential acceleration (\(a_t\)) is the linear acceleration at a point on a rotating object.

• Example: For a point 0.5 m from the axis with rad/s2, m/s2.

Torque

Torque (\(\tau\)) is the rotational equivalent of force, causing angular acceleration.

• Where r is the lever arm, F is the force, and \(\theta\) is the angle between them.

• Example: A 10 N force applied 0.3 m from the axis at 90°, N·m.

Center of Gravity

The center of gravity is the point where the total weight of an object acts.

• For uniform objects, it is at the geometric center.

• For irregular objects, it is found by balancing or calculation.

Moment of Inertia

Moment of inertia (\(I\)) measures an object's resistance to changes in rotational motion.

• (for discrete masses)

• Depends on mass distribution relative to the axis.

• Example: For a point mass, .

Newton's Second Law for Rotation

Relates torque, moment of inertia, and angular acceleration.

• Analogous to in linear motion.

• Example: If kg·m2 and rad/s2, N·m.

Chapter 8: Equilibrium and Elasticity

Static Equilibrium

An object is in static equilibrium if it is at rest and both the net force and net torque on it are zero.

• (translational equilibrium)

• (rotational equilibrium)

• Example: A balanced seesaw is in static equilibrium.

Stability

Stability depends on the position of the center of gravity relative to the base of support.

• Lower center of gravity and wider base increase stability.

• Objects tip when the center of gravity passes outside the base.

Hooke's Law

Hooke's law describes the force exerted by a spring when stretched or compressed.

• Where k is the spring constant, x is the displacement from equilibrium.

• Example: A spring with N/m stretched by 0.1 m exerts N.

Stress, Strain, and Young's Modulus

These concepts describe the elastic properties of materials.

• Stress: (force per unit area)

• Strain: (relative deformation)

• Young's modulus (Y):

• Example: A steel wire 2 m long stretches 1 mm under a 1000 N load. .

Chapter 9: Momentum and Impulse

Momentum and Impulse

Momentum (\(p\)) is the product of mass and velocity. Impulse is the change in momentum caused by a force acting over time.

• Example: A 2 kg object moving at 3 m/s has kg·m/s.

Impulse-Momentum Theorem

This theorem relates impulse to the change in momentum.

• Example: A 1 kg ball's velocity changes from 2 m/s to 5 m/s in 0.5 s. N.

Conservation of Momentum

In a closed system, total momentum before an event equals total momentum after.

• Applies to collisions and explosions.

• Example: Two carts collide and stick together; their combined momentum equals the sum of their initial momenta.

Collisions and Explosions

Collisions can be elastic (kinetic energy conserved) or inelastic (kinetic energy not conserved).

• In all collisions, momentum is conserved.

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

• In inelastic collisions, only momentum is conserved.

Angular Momentum and Its Conservation

Angular momentum (\(L\)) is the rotational analog of linear momentum. It is conserved in the absence of external torques.

• Conservation: if

• Example: A figure skater spins faster when pulling arms in, reducing and increasing to conserve .