CIRCULAR MOTION- A type of motion exhibited by bodies following a circular/curved path.

- Motion caused by centripetal force.

Related Terms:Rotation motion about an internal axisRevolution motion about an external axis The earth revolves around the sun. It takes earth a year to complete one revolution around the sun.To differentiate, for example: The earth rotates on its own axis. This is the reason why we are experiencing night and day.

Basic Rotational Quantities

The angular displacement is defined by:

For a circular path it follows that the angular velocity is

and the angular acceleration is

where the acceleration here is the tangential acceleration.

Basic Rotational Quantities

Angular velocity can be considered to be a vector quantity, with direction along the axis of rotation in the right-hand rule sense.

Angular velocityFor an object rotating about an axis, every point on the object has the same angular velocity. The tangential velocity of any point is proportional to its distance from the axis of rotation. Angular velocity has the units rad/s.

Angular velocity is the rate of change of angular displacement and can be described by the relationship

1. What is the angular velocity in rad/s of the second hand of a watch? Sample Problem:Answer: = 2rad / 60s = 0.10 rad/s2. A bicycle travels 141 m along a circular track of radius 15 m. What is the angular displacement in radians of the bicycle from its starting position?Answer: = 141m / 15m = 9.4 rad

Centripetal Force Any motion in a curved path represents accelerated motion, and requires a force directed toward the center of curvature of the path. This force is called the centripetal force which means "center seeking" force. where:m = mass of the object v = tangential velocityr = radius of the curved path

Centripetal Force The straight line motion in the absence of the constraining force (tension) is an example of Newtons first law of motion.

Centripetal Acceleration The centripetal acceleration can be derived for the case of circular motion since the curved path at any point can be extended to a circle.

Centripetal Acceleration

Problem 3: A 200g object is tied to the end of a cord and whirled in a horizontal circle of radius 1.20m at a constant 3.0 rev/s. Assume that the cord is horizontal, determine:(a) the acceleration of the object; and (b) the tension in the cord.

Solution 3:a. The object is not accelerating tangentially to the circle but is undergoing a radial or centripetal acceleration:where must be in rad/s, then, 3.0 rev/s = 6.0rad/s

b. To cause the acceleration found in (a), the cord must pull on the 0.20 kg mass with a centripetal force:This is the tension in the cord.

Centripetal force on banked highway curve The centripetal force is proportional to the square of the velocity, e.g. doubling of speed will require four times the centripetal force to keep the motion in a circle. If centripetal force must be provided by friction alone on a curve, an increase in speed could lead to an unexpected skid if friction is insufficient

Working Equations:Since centripetal force is provided by friction

If Ff = FN and FN = mg, then

The minimum speed required to make a turn

Also, the equation of = tan, therefore:where is the banking angle.

A car enters a horizontal, curved roadbed of radius 50 m. The coefficient of static friction between the tires and the roadbed is 0.20. What is the maximum speed with which the car can safely negotiate the unbanked curve?Problem 4:Solution:

Determine the minimum angle at which a roadbed should be banked so that a car traveling at 20m/s can safely negotiate the curve if the radius of the curve is 200m.Problem 5:Solution:

Motion in a Vertical Circle

Motion in a Vertical Circle The motion of a mass on a string in a vertical circle includes a number of mechanical concepts. 1. It must satisfy the constraints of centripetal force to remain in a circle,2. It must satisfy the demands of conservation of energy as gravitational potential energy is converted to kinetic energy when the mass moves downward. 3. The velocity must increase as the mass moves downward from the top of the circle, subject to the constraints stated.

Motion in a Vertical Circle Using the centrifugal force conditions, the tension at the bottom can be related to the tension at the top:

A 0.75-kg ball is attached to a 1.0-m rope and whirled in a vertical circle. The rope will break when the tension exceeds 450 N. What is the maximum speed the ball can have at the bottom of the circle without breaking the rope?

Problem 6:Solution:

Circular Orbit

The force of gravity in keeping an object in circular motion is an example of centripetal force. Since it acts always perpendicular to the motion, gravity does not do work on the orbiting object if it is in a circular orbit.

What is the acceleration due to gravity at an altitude of 1.00 106 m above the earth's surface? (Note: the radius of the earth is 6.38 106 m.)

Problem 7:Solution:

Assignment:In an amusement park ride, a small child stands against the wall of a cylindrical room that is then made to rotate. The floor drops downward and the child remains pinned against the wall. If the radius of the device is 2.15 m and the relevant coefficient of friction between the child and the wall is 0.400, with what minimum speed is the child moving if he is to remain pinned against the wall?

2. An indoor track is to be designed such that each end is a banked semi-circle with a radius of 24 m. What should the banking angle be for a person running at speed v = 6.0 m/s?

3. Outdoor Activity in Motion- tell your experience about motion in an amusement park.- include your picture in motion- written output must be 1-2 pages only and must be compiled by group (lab. group) in a clear book- deadline: before midterm exam- other option: experience motion within your vicinity