Linear Momentum, Collisions, and Rotational Dynamics: Study Notes

Linear Momentum

Definition and Properties

Linear momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is a vector quantity, meaning it has both magnitude and direction.

• Definition: The linear momentum p of an object is defined as the product of its mass m and velocity v.

• Formula: $\vec{p} = m\vec{v}$

• Conservation of Momentum: In a closed system with no external forces, the total linear momentum remains constant.

• Impulse: The change in momentum of an object is called impulse, given by the product of force and the time interval over which it acts. $\vec{J} = \Delta \vec{p} = \vec{F} \Delta t$

• Example: A 2 kg ball moving at 3 m/s has a momentum of $2 \times 3 = 6$ kg·m/s.

Collisions

Inelastic Collisions

In inelastic collisions, objects collide and may stick together, resulting in a loss of kinetic energy, though momentum is conserved.

• Perfectly Inelastic Collision: The colliding objects stick together after impact.

• Conservation of Momentum: $m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f$

• Kinetic Energy: Not conserved; some is transformed into other forms (e.g., heat, deformation).

• Example: Two cars of masses 1000 kg and 800 kg, moving at 10 m/s and 0 m/s respectively, stick together after collision. Final velocity: $v_f = \frac{1000 \times 10 + 800 \times 0}{1000 + 800} = 5.56$ m/s

Elastic Collisions

Elastic collisions are those in which both momentum and kinetic energy are conserved.

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

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

• Example: Two billiard balls of equal mass collide head-on; their velocities are exchanged after the collision.

Rotational Kinematics

Angular Position, Velocity, and Acceleration

Rotational kinematics describes the motion of objects as they rotate about an axis.

• Angular Position (θ): The angle an object has rotated, measured in radians.

• Angular Velocity (ω): The rate of change of angular position. $\omega = \frac{d\theta}{dt}$

• Angular Acceleration (α): The rate of change of angular velocity. $\alpha = \frac{d\omega}{dt}$

• Example: A wheel rotates 2 radians in 1 second; its angular velocity is 2 rad/s.

Rotational Kinematics Equations

These equations are analogous to linear kinematics, describing rotational motion under constant angular acceleration.

• $\omega_f = \omega_i + \alpha t$

• $\theta = \omega_i t + \frac{1}{2} \alpha t^2$

• $\omega_f^2 = \omega_i^2 + 2\alpha \theta$

• Example: If a disk starts from rest and accelerates at 3 rad/s² for 4 seconds, its final angular velocity is $0 + 3 \times 4 = 12$ rad/s.

Connections Between Linear and Rotational Quantities

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

• Linear Displacement: $s = r\theta$

• Linear Velocity: $v = r\omega$

• Linear Acceleration: $a = r\alpha$

• Example: A point on a wheel of radius 0.5 m rotating at 4 rad/s has a linear velocity of $0.5 \times 4 = 2$ m/s.

Torque

Definition and Calculation

Torque is the rotational equivalent of force, causing objects to rotate about an axis.

• Definition: Torque (τ) is the product of force and the lever arm (distance from axis of rotation).

• Formula: $\tau = r F \sin \theta$

• Direction: Determined by the right-hand rule.

• Example: A force of 10 N applied perpendicular to a lever 0.3 m from the pivot produces a torque of $0.3 \times 10 = 3$ N·m.

Torque and Angular Acceleration

Torque causes angular acceleration in a rotating object, analogous to how force causes linear acceleration.

• Newton's Second Law for Rotation: $\tau_{net} = I \alpha$

• Moment of Inertia (I): Measures an object's resistance to changes in rotational motion.

• Example: A disk with $I = 2$ kg·m² and net torque of 6 N·m has angular acceleration $\alpha = 6 / 2 = 3$ rad/s².

Dynamic Applications of Torque

Torque is essential in analyzing the rotational dynamics of objects, including machinery, vehicles, and physical systems.

• Balancing Forces: In static equilibrium, the sum of torques about any axis is zero.

• Rotational Dynamics: Used to solve problems involving angular acceleration, rotational energy, and mechanical advantage.

• Example: Calculating the torque required to accelerate a flywheel in an engine.

Quantity	Linear	Rotational
Displacement	$x$	$\theta$
Velocity	$v$	$\omega$
Acceleration	$a$	$\alpha$
Mass / Moment of Inertia	$m$	$I$
Force / Torque	$F$	$\tau$

Additional info: Some context and examples have been inferred to provide a complete, self-contained study guide based on the listed textbook sections and fragmented notes.