Physics with Calculus: Exam 2 Study Guide (Chapters 5–9)

Ch. 5: Centripetal Motion and Non-Uniform Circular Motion

Centripetal Motion

Centripetal motion refers to the movement of an object along a circular path, where a net force acts toward the center of the circle, keeping the object in motion. This force can arise from various sources:

• Tension: The force exerted by a string or rope when an object is swung in a circle.

• Normal Force: The perpendicular contact force from a surface, such as a car turning on a flat road.

• Gravity: The force responsible for planetary orbits and satellite motion.

• Friction: The force between tires and road, allowing vehicles to turn without slipping.

• Banking: The tilt of a road or track to help vehicles negotiate curves safely.

The magnitude of the required centripetal force is given by:

$F_c = \frac{mv^2}{r}$

where m is mass, v is velocity, and r is radius of the circle.

Non-Uniform Circular Motion

Non-uniform circular motion occurs when the speed of the object changes as it moves along the circular path, resulting in both radial (centripetal) and tangential acceleration.

• Radial Acceleration: Directed toward the center, responsible for changing direction.

• Tangential Acceleration: Responsible for changing the speed along the path.

$a_{rad} = \frac{v^2}{r}$ $a_{tan} = \frac{d|v|}{dt}$

Ch. 6: Gravitation and Orbital Mechanics

Universal Law of Gravitation

Newton's Universal Law of Gravitation describes the attractive force between two masses:

$F = G\frac{m_1 m_2}{r^2}$

where G is the gravitational constant, m_1 and m_2 are masses, and r is the distance between centers.

Surface Gravity

Surface gravity is the acceleration due to gravity experienced at the surface of a planet or celestial body:

$g = G\frac{M}{R^2}$

where M is the mass and R is the radius of the body.

Orbital Mechanics

Orbital mechanics studies the motion of objects under the influence of gravity, including satellites and planets. Key concepts include:

• Kepler's Laws: Describe planetary motion.

• Orbital Velocity: $v = \sqrt{G\frac{M}{r}}$

• Period of Orbit: $T = 2\pi\sqrt{\frac{r^3}{GM}}$

Ch. 7: Work and Energy

Work Done by a Force

Work is the energy transferred by a force acting over a distance:

$W = Fd\cos\theta$

where F is force, d is displacement, and \theta is the angle between force and displacement.

Work Done by a Varying Force

For a varying force, work is calculated using the integral:

$W = \int_{a}^{b} F(x) dx$

Dot Product (Scalar Product)

The dot product is used to calculate work when force and displacement are vectors:

$\vec{A} \cdot \vec{B} = |A||B|\cos\theta$

Work-Energy Theorem

The work-energy theorem states that the net work done on an object equals its change in kinetic energy:

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

Ch. 8: Potential Energy and Conservation of Energy

Potential Energies

Potential energy is stored energy due to position or configuration:

• Gravitational Potential Energy: $U_g = mgh$ (near Earth's surface), $U_g = -G\frac{m_1 m_2}{r}$ (universal form)

• Spring Potential Energy: $U_s = \frac{1}{2}kx^2$

Deriving Potential Energy from a Force

Potential energy can be derived from a conservative force:

$U(x) = -\int F(x) dx$

Deriving a Force from Potential Energy

The force associated with a potential energy function is:

$F(x) = -\frac{dU}{dx}$

Conservation of Energy

The total energy in a closed system remains constant:

$E_{total} = K + U$

where K is kinetic energy and U is potential energy.

• Dissipative Forces: Forces like friction convert mechanical energy to thermal energy.

• Escape Velocity: The minimum speed needed to escape a gravitational field: $v_{esc} = \sqrt{2G\frac{M}{R}}$

Power

Power is the rate at which work is done:

$P = \frac{dW}{dt}$

For constant force and velocity: $P = Fv$

Ch. 9: Impulse, Momentum, and Collisions

Impulse

Impulse is the change in momentum produced by a force acting over time:

$J = F\Delta t = \Delta p$

Conservation of Momentum

In a closed system, total momentum is conserved:

$p_{total, initial} = p_{total, final}$

Types of Collisions

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

• Inelastic Collisions: Momentum is conserved, but kinetic energy is not.

• Completely Inelastic Collisions: Colliding objects stick together after impact.

Ballistic Pendulum Problems

Ballistic pendulum problems involve a projectile embedding in a pendulum, used to measure velocity via conservation of momentum and energy.

2D Collisions

Collisions in two dimensions require conservation of momentum in both x and y directions:

$m_1 v_{1x} + m_2 v_{2x} = m_1 v'_{1x} + m_2 v'_{2x}$ $m_1 v_{1y} + m_2 v_{2y} = m_1 v'_{1y} + m_2 v'_{2y}$

Center of Mass

The center of mass is the weighted average position of all mass in a system:

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

It determines the motion of the system as a whole.