Circular Motion

Chapter 9

Circular Motion

• Axis – is the straight line around which rotation takes place.

• Internal Axis - is located within the body of the object.

Circular Motion

• Rotation – is the spin around the internal axis.

• External axis – is outside the body of the object.

Circular Motion

• Linear speed - is distance/time

• Since the outer edge of on object moving in a circle moves further it has greater linear speed.

Circular Motion

• Tangential speed – is the same as linear speed only with a circular motion.

Circular Motion

• Rotational speed - is the # of rotations/time

• Tangential speed is approximately equal to radial distance x rotational speed.

Circular Motion

• IN ANY RIGIDLY ROTATING SYSTEM, ALL PARTS HAVE THE SAME ROTATIONAL SPEED.

Circular Motion

• Period (T) is the time it takes for one full rotation or revolution of an object. (measured in seconds)

Circular Motion

• Frequency (f) is the # of rotations or revolutions per unit of time. (measured in Hertz Hz).

Circular Motion

• T = 1/f

• f = 1/T

Circular Motion

• Every object exerts gravitational force on every other object.

• The force depends on how much mass the objects have and on how far apart they are.

• The force is hard to detect unless at least one of the objects has a lot of mass.

Circular Motion

• Gravity is the force that keeps planets in orbit around the sun and governs the motion of the rest of the solar system.

• Gravity alone holds us to the earth’s surface and explains the phenomenon of the tides.

Reminders

• Velocity is speed and the direction of travel.

• Acceleration is the rate of change of velocity.

• Force cause the acceleration of motion.

• Work is done on an object to change the energy of the object.

Some definitions

• Centripetal means “center seeking”

• Centrifugal means “center fleeing”

Circular Motion

• Consider a Ferris wheel. The cars on the rotating Ferris wheel are said to be in circular motion.

Circular Motion

• Any object that revolves about a single axis undergoes circular motion.

• The line about which the rotation occurs is called the axis of rotation.

• In this case, it is a line perpendicular to the side of the Ferris wheel and passing through the wheel’s center.

Tangential speed

• Tangential speed (vi) can be used to describe the speed of an object in circular motion.

• The tangential speed of a car on the Ferris wheel is the car’s speed along an imaginary line drawn tangent to the car’s circular path.

Tangential speed

• This definition can be applied to an object moving in circular motion.

• When the tangential speed is constant, the motion is described as uniform circular motion.

Tangential speed

• The tangential speed depends on the distance from the object to the center of the circular path.

Tangential speed

• For example, consider a pair of horses side-by-side on a carousel.

• Each completes one full circle in the same time period, but the horse on the outside covers more distance than the inside horse does, so the outside horse has a greater tangential speed.

Centripetal Acceleration

• Suppose a car on a Ferris wheel is moving at a constant speed around the wheel.

• Even though the tangential speed is constant, the car still has an acceleration.

Centripetal Acceleration

• a = vf - vi

tf - ti

________

Centripetal Acceleration

• Or put in words, centripetal acceleration is equal to linear speed squared divided by the radius.

Speed

• V = (2Πr) / T

Centripetal Acceleration

• Acceleration depends on a change in the velocity.

• Because velocity is a vector, acceleration can be produced by a change in the magnitude of the velocity, a change in the direction of the velocity, or both.

Centripetal Acceleration

• The acceleration of a Ferris wheel car moving in a circular path and at constant speed is due to a change in direction.

• An acceleration of this nature is called a centripetal acceleration.

Centripetal Acceleration

• The magnitude of a centripetal acceleration is given by the following equation.

• Centripetal acceleration = (tangential speed)2 / radius of circular path

Example

• A test car moves at a constant speed around a circular track. If the car is 48.2 meters from the track’s center and has a centripetal acceleration of 8.05 m/s2 what is the car’s tangential speed?

Solution

• Given:

r = 48.2 m

ac = 8.05 m/s2

Unknown:

vt = ?

Solution

• ac = vt2 / r

• 8.05 m/s2 = vt2 / 48.2 m

• vt = 19.7 m/s