Physics 220 Exam 3 Review: Work, Energy, Momentum, and Rotational Motion

Work, Energy, and Power

Definitions and Concepts

Understanding the concepts of work, kinetic energy, and potential energy is fundamental to analyzing physical systems in mechanics.

• Work (W): The energy transferred to or from an object via the application of force along a displacement. Defined as:

$W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

• Kinetic Energy (K): The energy of motion, given by:

$K = \frac{1}{2}mv^2$

• Potential Energy (U): The energy stored due to an object's position or configuration.

  • Gravitational Potential Energy: $U_g = mgh$

  • Elastic Potential Energy (spring): $U_e = \frac{1}{2}kx^2$

Calculating Work

• Constant Force: Use $W = Fd\cos\theta$ to determine the work done by a constant force. The sign of work depends on the angle between force and displacement.

• Varying Force: For a force that changes with position, work is calculated as:

$W = \int_{x_i}^{x_f} F(x)\,dx$

• Elastic Force: The force exerted by a spring is $F = -kx$ (Hooke's Law).

Work-Energy Theorem

• The net work done on an object equals the change in its kinetic energy:

$W_{\text{net}} = \Delta K = K_f - K_i$

• This theorem can be used to find changes in speed, given the total work done.

Conservative and Non-Conservative Forces

• Conservative Forces: Work done is path-independent (e.g., gravity, spring force).

• Non-Conservative Forces: Work done depends on the path (e.g., friction).

Conservation of Energy

• In the absence of non-conservative forces, the total mechanical energy (kinetic + potential) is conserved:

$K_i + U_i = K_f + U_f$

• Use this principle to solve for unknowns such as speed, height, or displacement.

Momentum and Impulse

Definitions

• Momentum (\(\vec{p}\)): The product of mass and velocity:

$\vec{p} = m\vec{v}$

• Impulse (\(\vec{J}\)): The change in momentum, equal to the net force applied over a time interval:

$\vec{J} = \Delta \vec{p} = \int \vec{F}\,dt$

Conservation of Momentum

• Momentum is conserved in a system if the net external force is zero:

$\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}$

• Applies to both one-dimensional and two-dimensional collisions.

Collisions

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

• Inelastic Collisions: Momentum is conserved, but kinetic energy is not. In a perfectly inelastic collision, objects stick together after the collision.

• To find velocities before or after collisions, apply conservation laws and solve the resulting equations.

Impulse-Momentum Theorem

• Relates impulse to the change in momentum and the net force:

$\vec{J} = \Delta \vec{p} = \vec{F}_{\text{net}} \Delta t$

Center of Mass

• The point representing the average position of the mass distribution of a system.

$x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}$

• Important for analyzing motion of extended objects and systems.

Rotational Motion

Angular Kinematic Quantities

• Angular Position (\(\theta\)): The angle describing the rotation of an object.

• Angular Velocity (\(\omega\)): The rate of change of angular position:

$\omega = \frac{d\theta}{dt}$

• Angular Acceleration (\(\alpha\)): The rate of change of angular velocity:

$\alpha = \frac{d\omega}{dt}$

Rotational Kinematics with Constant Angular Acceleration

• Equations analogous to linear kinematics:

$\omega_f = \omega_i + \alpha t$

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

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

Relating Angular and Linear Quantities

• For a point at a distance r from the axis of rotation:

  • Arc length: $s = r\theta$

  • Tangential velocity: $v_{\text{tan}} = r\omega$

  • Tangential acceleration: $a_{\text{tan}} = r\alpha$

  • Radial (centripetal) acceleration: $a_{\text{rad}} = r\omega^2$

Rotational Kinetic Energy and Moment of Inertia

• Rotational Kinetic Energy:

$K_{\text{rot}} = \frac{1}{2}I\omega^2$

• Moment of Inertia (I): The rotational analog of mass, depends on mass distribution:

$I = \sum m_i r_i^2$

Rolling Without Slipping

• Occurs when the point of contact between a rolling object and the surface is instantaneously at rest relative to the surface.

• Condition: $v_{\text{cm}} = R\omega$

• Total kinetic energy of a rolling object:

$K_{\text{total}} = \frac{1}{2}mv_{\text{cm}}^2 + \frac{1}{2}I\omega^2$

Conservation of Mechanical Energy in Rotational Systems

• For systems involving rolling or spinning objects, mechanical energy conservation includes both translational and rotational kinetic energy:

$K_{\text{trans}} + K_{\text{rot}} + U = \text{constant}$

• Use this to solve for unknowns in problems involving rolling motion.

Example: Rolling Sphere Down an Incline

• A solid sphere of mass m and radius R rolls without slipping down a height h. Find its speed at the bottom.

• Apply conservation of energy:

$mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

• For a solid sphere, $I = \frac{2}{5}mR^2$ and $v = R\omega$.

• Solve for v:

$v = \sqrt{\frac{10}{7}gh}$

Additional info: The above example illustrates the application of energy conservation to rolling motion, combining both translational and rotational kinetic energy.