5.1 Uniform Circular Motion

• Uniform circular motion - the motion of an object traveling at a constant speed on a circular path (constant speed = uniform).

  • Equation for circular velocity: v_{c}=\dfrac{2\pi r}{T}

  • Speed (magnitude) is constant

  • Direction is not constant, so velocity is not constant.

  • Vector quantity

  • SI Units: m/s

• Revolution - a singular circular orbit of one body around a point.

• Period - the time (T) it takes an object to travel once around a circle (to make one revolution).

  • Period (T)=\dfrac{1}{frequency}

  • Scalar quantity

  • SI Unit: second (s)

• Equation for circumference of a circle: 2\pi r

  • SI Unit: m

5.2 Centripetal Acceleration

• In uniform circular motion, speed is constant, but direction of the velocity vector is not constant.

  When an object's speed is constant but its velocity vector is not, it's moving in a circular motion. This is because the object's direction is constantly changing, even though its speed remains constant.

• Centripetal acceleration (a_{c}) directed towards the center of the circle; the same direction as the change is velocity

  • Centripetal acceleration is center seeking.

  • a_{c}=\dfrac{v^{2}}{r}

  • Vector quantity

  • SI Units: m/s²

• Spinning an object in circular motion

  • When the object is released, it will move along a straight path.

  • This is due to Newton’s First Law - an object in motion stays in motion unless acted on by a force.

  • Once the object is released, there is no net force acting on it.

5.3 Centripetal Force

• Centripetal force (F_{c}) The net force required to produce centripetal acceleration to keep an object moving in a circular path (newton’s second law).

• Direction: ALWAYS towards the center of the circle, continually changes direction as the object moves.

• There are two ways to write this equation:

  • F_{c}=ma_{c}

  • F_{c}=\dfrac{mv^{2}}{r}

    • The second equation inserts the centripetal acceleration equation.

  • SI Units: Newton (N)

5.5 Satellites in Circular Orbits

• There is only one speed and period that a satellite can have if the satellite is to remain in an orbit with a fixed radius.

• When something is in orbit, gravity is the centripetal force.

• The speed of a satellite:

  • v=\sqrt{\dfrac{GM_{E}}{r}}

• The period of a satellite:

  • T=\dfrac{2\pi r^{\dfrac{3}{2}}}{\sqrt{GM_{E}}}

• Universal gravitation constant: G = 6.67\times 10^{-11}\dfrac{Nm^{2}}{kg^{2}}

5.6 Apparent Weightlessness and Artificial Gravity

• Weight is not the same as mass. Weight is the force of gravity working on you.

• Apparent Weightlessness - apparent weightlessness occurs when an object is falling freely under the influence of gravity, giving the sensation of weightlessness.

• Artificial Gravity - Artificial gravity is a concept of simulating gravity through centrifugal force or acceleration to create a gravitational effect in space or other environments.

Extra Notes

Circular motion refers to the movement of an object along a circular path. It involves two key concepts: centripetal force and centripetal acceleration.

1. Centripetal force: It is the force that acts towards the center of the circular path, keeping the object in motion. It is responsible for changing the direction of the object's velocity but not its magnitude.

2. Centripetal acceleration: It is the acceleration experienced by an object moving in a circular path. It is always directed towards the center of the circle and is perpendicular to the object's velocity.

Some key principles of circular motion include:

• The speed of an object in circular motion remains constant if there is no external force acting on it.

• The centripetal force required for circular motion depends on the mass of the object, its speed, and the radius of the circular path.

• The centripetal force can be provided by various factors, such as tension in a string, gravitational force, or friction.

Understanding circular motion is crucial in various fields, including physics, engineering, and astronomy, as it helps explain phenomena like planetary orbits, roller coasters, and the motion of satellites.