Circular Motion, Gravitation, Work, and Energy – Study Notes for College Physics

Circular Motion & Gravitation

Forces in Circular Motion

Circular motion occurs when an object moves along a circular path. The force that keeps an object moving in a circle is called the centripetal force. This force is always directed toward the center of the circle and is responsible for changing the direction of the object's velocity, not its speed.

• Centripetal Acceleration: The acceleration directed toward the center of the circle, given by $a_c = \frac{v^2}{r}$, where $v$ is the speed and $r$ is the radius of the circle.

• Centripetal Force: The net force causing the centripetal acceleration, $F_c = m a_c = m \frac{v^2}{r}$.

• Examples: Planets orbiting the sun, satellites orbiting Earth, cars turning on a curved road.

Motion in a Vertical Circle

When an object moves in a vertical circle, such as a pendulum or a roller coaster loop, the forces acting on it include gravity and the tension or normal force from the path. The net force at any point is the sum of these forces, and the speed varies due to changes in gravitational potential energy.

Newton’s Law of Gravitation

Newton’s Law of Universal Gravitation states that every two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers:

• $F = G \frac{m_1 m_2}{r^2}$, where $G$ is the gravitational constant ($6.674 \times 10^{-11} \ \mathrm{N \cdot m^2 / kg^2}$).

• This law explains planetary motion, satellite orbits, and the behavior of objects near Earth's surface.

Weight and Its Variation with Altitude

The weight of an object is the gravitational force exerted on it by Earth. Weight decreases with altitude because the distance from Earth's center increases, reducing the gravitational force:

• $w = G \frac{m_E m}{r^2}$, where $m_E$ is Earth's mass, $m$ is the object's mass, and $r$ is the distance from Earth's center.

• As altitude increases, $r$ increases, so $w$ decreases.

Earth’s Density Profile

Earth is not uniform in density; its density decreases with distance from the center. The core is denser than the mantle and crust.

• Solid inner core: Highest density.

• Molten outer core: Lower density than inner core.

• Mostly solid mantle: Density decreases outward.

Satellite Motion

Satellites in orbit around Earth follow paths determined by their initial speed and the gravitational pull of Earth. For a circular orbit, the satellite's speed is just right to keep its distance from Earth's center constant. The gravitational force provides the necessary centripetal acceleration.

• Circular Orbit Speed: $v = \sqrt{\frac{G m_E}{r}}$

• Applications: International Space Station, GPS satellites.

Black Holes

A black hole is a region in space where gravity is so strong that not even light can escape. Black holes are detected by observing x-rays emitted from the accretion disks of matter spiraling into them.

• Schwarzschild radius ($R_s$): The radius at which the escape velocity equals the speed of light.

• Accretion disk: Matter from a companion star forms a disk around the black hole, emitting x-rays as it heats up.

Work and Energy

Overview of Energy

Energy is the ability to do work. It exists in various forms, including kinetic, potential, thermal, chemical, and nuclear energy. Energy is a conserved quantity, meaning it cannot be created or destroyed, only transformed from one form to another.

• Kinetic Energy: Energy of motion.

• Potential Energy: Stored energy due to position or configuration.

• Conservation of Energy: The total energy of an isolated system remains constant.

Work

Work is done when a force causes a displacement. The amount of work depends on the magnitude of the force, the displacement, and the angle between them:

• $W = F s \cos \phi$

• Alternatively, $W = \vec{F} \cdot \vec{s}$ (dot product of force and displacement vectors).

Example: Work Done by a Force

Steve pushes a stalled car 19 m with a force of 210 N. If he pushes in the direction of motion, $\phi = 0$, so $W = (210\,\mathrm{N})(19\,\mathrm{m}) = 4.0 \times 10^3\,\mathrm{J}$. If he pushes at $30^\circ$, $W = (210\,\mathrm{N})(0.866)(19\,\mathrm{m}) = 3.5 \times 10^3\,\mathrm{J}$.

Direction of Force and Work

The sign of work depends on the direction of the force relative to displacement:

• Positive work: Force and displacement in the same direction.

• Negative work: Force opposes displacement.

• Zero work: Force is perpendicular to displacement or displacement is zero.

Work Done by Several Forces

When multiple forces act on an object, the total work is the sum of the work done by each force. For example, a tractor pulls a sled while friction opposes the motion. The net work is the algebraic sum of the work done by all forces.

Kinetic Energy

Kinetic energy is the energy of motion, given by:

• $K = \frac{1}{2} m v^2$

• Measured in joules (J).

• Depends on mass and the square of speed, not direction.

Work-Energy Theorem

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

• $W_{\text{tot}} = K_2 - K_1 = \Delta K$

• If $W_{\text{tot}} > 0$, the particle speeds up; if $W_{\text{tot}} < 0$, it slows down.

Work Done by a Varying Force

When the force varies with position, the work done is equal to the area under the force vs. position graph:

• For a constant force, $W = F s$ (area of a rectangle).

• For a varying force, sum the areas under the curve (can use calculus for continuous functions).

Work Done on a Spring (Hooke’s Law)

The force required to stretch or compress a spring is proportional to the displacement, described by Hooke’s Law: $F_x = -k x$. The work done in stretching or compressing a spring is:

• $W = \frac{1}{2} k x^2$

Potential Energy

Potential energy is energy stored due to an object's position or configuration. Gravitational potential energy near Earth's surface is given by:

• $U = m g y$

• Change in potential energy: $\Delta U = m g (y_2 - y_1)$

Conservation of Mechanical Energy

When only conservative forces (like gravity) do work, the total mechanical energy (kinetic + potential) of a system remains constant:

• $E = K + U = \text{constant}$

• Applies to free-falling objects, pendulums, and roller coasters (ignoring friction).

Work and Energy Along a Curved Path

The work done by gravity depends only on the vertical displacement, not the path taken. The same expression for gravitational potential energy applies whether the path is straight or curved:

• $W_{\text{grav}} = m g (y_1 - y_2)$

• Horizontal motion does not affect the work done by gravity.