Unit 3: Circular Motion and Gravitation

# Uniform Circular Motion

Uniform circular motion is the motion of an object moving in a circular path at a constant speed. In this type of motion, the object's velocity is constantly changing due to the change in direction of its motion. You should remember that although the speed may be constant, the velocity is not because the direction is always changing meaning that the velocity is always changing. Since the velocity is changing, there must be acceleration. The acceleration does not change the speed of the object, rather it changes the direction of the velocity to keep the object moving along the circular path.

## Centripetal Force and Acceleration

Centripetal force is the force that acts on an object moving in a circular path, directed towards the center of the circle. It is responsible for keeping the object moving in a circular path. The centripetal acceleration is what turns the velocity vectors to keep the object traveling in a circle. The magnitude of the centripetal acceleration depends on the object’s speed, v, and the radius of the circular path, r,

`a꜀ = v^2/r`

where a꜀ is the centripetal acceleration, v is the velocity of the object, and r is the radius of the circle.

The centripetal force required to keep an object moving in a circular path is given by the formula:

F = ma = mv^2 / r

where F is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circle.

Some examples of uniform circular motion include the motion of a car around a circular track, the motion of a satellite orbiting the Earth, and the motion of a ball on a string being swung in a circle.

# Gravitational and Electric Forces

The four fundamental forces in physics are the **gravitational force**, the **electromagnetic force**, the **weak force**, and the **strong force**. **Gravity dominates** at the largest mass and distance scales because its effects are proportional to the mass of the objects involved.

## Gravitational Force

The gravitational force is the force of attraction between two masses. It is one of the weakest forces in nature, but it dominated at large mass and distances. An example of this is our solar system where the huge gravitational pull of the sun keep the planets in orbit.

Attractive force between two objects with mass

Proportional to the product of their masses and inversely proportional to the square of the distance between them

Described by Newton's Law of Universal Gravitation: F = G * (m1 * m2) / r^2

G is the gravitational constant (6.674 * 10^-11 N * m^2 / kg^2)

## Electric Force

Attractive or repulsive force between two charged objects

Proportional to the product of their charges and inversely proportional to the square of the distance between them

Described by Coulomb's Law: F = k * (q1 * q2) / r^2

k is the Coulomb constant (9 * 10^9 N * m^2 / C^2)

# Gravitational Acceleration

Gravitational acceleration is the acceleration experienced by an object due to the force of gravity. It is denoted by the symbol 'g' and is measured in meters per second squared (m/s^2). The gravitational force is always acting **vertically downward** towards the center of a planet. If this is the only force that is being exerted on an object at a specific time, then the object is considered to be in **free fall**.

## Formula

The formula for gravitational acceleration is:

g = G * M / r^2

where:

G is the gravitational constant (6.674 * 10^-11 N * m^2 / kg^2)

M is the mass of the object causing the gravitational force

r is the distance between the object and the center of mass of the other object

On Earth, the value of gravitational acceleration is approximately 9.81 m/s^2. This means that an object in free fall near the surface of the Earth will accelerate at a rate of 9.81 m/s^2. The variable g, or the gravitational field is subject to change based on the planet itself or the object’s location relative to the planet’s surface. On Earth, we know g as 9.8 m/s^2, but on other planets, the value is different.

`F₉ = m1a`

Gm1m2/r^2 = m1a

a = gm/r^2

g = Gm/r^2

# Unit 3: Circular Motion and Gravitation

# Uniform Circular Motion

Uniform circular motion is the motion of an object moving in a circular path at a constant speed. In this type of motion, the object's velocity is constantly changing due to the change in direction of its motion. You should remember that although the speed may be constant, the velocity is not because the direction is always changing meaning that the velocity is always changing. Since the velocity is changing, there must be acceleration. The acceleration does not change the speed of the object, rather it changes the direction of the velocity to keep the object moving along the circular path.

## Centripetal Force and Acceleration

Centripetal force is the force that acts on an object moving in a circular path, directed towards the center of the circle. It is responsible for keeping the object moving in a circular path. The centripetal acceleration is what turns the velocity vectors to keep the object traveling in a circle. The magnitude of the centripetal acceleration depends on the object’s speed, v, and the radius of the circular path, r,

`a꜀ = v^2/r`

where a꜀ is the centripetal acceleration, v is the velocity of the object, and r is the radius of the circle.

The centripetal force required to keep an object moving in a circular path is given by the formula:

F = ma = mv^2 / r

where F is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circle.

Some examples of uniform circular motion include the motion of a car around a circular track, the motion of a satellite orbiting the Earth, and the motion of a ball on a string being swung in a circle.

# Gravitational and Electric Forces

The four fundamental forces in physics are the **gravitational force**, the **electromagnetic force**, the **weak force**, and the **strong force**. **Gravity dominates** at the largest mass and distance scales because its effects are proportional to the mass of the objects involved.

## Gravitational Force

The gravitational force is the force of attraction between two masses. It is one of the weakest forces in nature, but it dominated at large mass and distances. An example of this is our solar system where the huge gravitational pull of the sun keep the planets in orbit.

Attractive force between two objects with mass

Proportional to the product of their masses and inversely proportional to the square of the distance between them

Described by Newton's Law of Universal Gravitation: F = G * (m1 * m2) / r^2

G is the gravitational constant (6.674 * 10^-11 N * m^2 / kg^2)

## Electric Force

Attractive or repulsive force between two charged objects

Proportional to the product of their charges and inversely proportional to the square of the distance between them

Described by Coulomb's Law: F = k * (q1 * q2) / r^2

k is the Coulomb constant (9 * 10^9 N * m^2 / C^2)

# Gravitational Acceleration

Gravitational acceleration is the acceleration experienced by an object due to the force of gravity. It is denoted by the symbol 'g' and is measured in meters per second squared (m/s^2). The gravitational force is always acting **vertically downward** towards the center of a planet. If this is the only force that is being exerted on an object at a specific time, then the object is considered to be in **free fall**.

## Formula

The formula for gravitational acceleration is:

g = G * M / r^2

where:

G is the gravitational constant (6.674 * 10^-11 N * m^2 / kg^2)

M is the mass of the object causing the gravitational force

r is the distance between the object and the center of mass of the other object

On Earth, the value of gravitational acceleration is approximately 9.81 m/s^2. This means that an object in free fall near the surface of the Earth will accelerate at a rate of 9.81 m/s^2. The variable g, or the gravitational field is subject to change based on the planet itself or the object’s location relative to the planet’s surface. On Earth, we know g as 9.8 m/s^2, but on other planets, the value is different.

`F₉ = m1a`

Gm1m2/r^2 = m1a

a = gm/r^2

g = Gm/r^2