Uniform Circular Motion Physics

What is Uniform Circular Motion? Velocity exists not only in linear equations, but also in circular paths. Objects can rotate and revolve. A good example of this is Earth.

Rotation Rotation: When an object spins or rotates around an internal axis. Example: It takes Earth 24 hours to rotate on it’s axis.

Revolution Revolution: when an object turns or revolves around an external axis. Example: It takes Earth 365 days to revolve around the sun.

Period: the time (T) to complete one cycle of motion Period: the time (T) to complete one cycle of motion. (rotation or revolution) Frequency: the number of cycles of motion (revolutions or rotations) in one second. (measured in Hz) – Also, angular speed (w):

Linear speed and tangential speed linear speed: distance moved per unit of time (speed). Ex: Merry Go Round tangential speed (Vt) : the speed of an object that is moving along a circular path. The direction of motion is tangent to the circle.

When dealing with motion in a circular path, velocity changes are really direction changes, and acceleration changes are equal to force.

Centripetal Force: this causes circular motion, and is directed towards the center of the circle. Example of centripetal force: the moon being held in orbit by gravity. Centrifugal Force: an imaginary force that seems to pull away from the center of the circle which is caused by inertia. Example of centrifugal force: carnival rides.

Key Points for Uniform Circular Motion All points on a rotating rigid object will have the same angular speed (and thus the same angular acceleration). The tangential speed of an object placed on a rotating body will increase as it is moved away from the center of the rigid body. On object that is following a circular path has a net force (Fc) and acceleration (ac) that are acting towards the center of the object. (Think Geometry)

Key points continued Velocity is always tangent to the circle and is always changing. Acceleration is towards the center of the circle Look at the following rotating disk. Which point has the greater tangential velocity? Which point has the greater angular velocity?

Equations: V=2πr/T a= v2/r or 4∏2r/T2 So….F = m x v2 / r F= (m)(4π2)(r) T2

Sample Problem I What is the frequency and period of a bug that makes 5 revolutions on a DVD in 2 seconds?

Sample Problem II What is the tangential speed of a dude rotating at a frequency of 5 Hz while sitting 3 m from the center of a carousel? If the dude’s magical unicorn where to instantly disappear, what direction would he fly?

Sample Problem III Some random guy is twirling his grand-fathers 2 kg time piece which is tied to the end of a .8 m chain. The time piece travels at a frequency of 3 revolutions every 2 seconds. What is the cetripetal acceleration of the time piece? What is the Tension (Centriptal FORCE) in the chain?

Angular speed, force, period angular speed (w): the number of rotations per unit of time. Also called rotational speed or the objects FREQUENCY! centripetal force (Fc): any force that will cause an object to take a circular path.

Tension, Normal Force & Circular Motion Remember: Velocity is always tangent to the circle and is always changing. Acceleration goes towards the center of the circle ∑F’s are written Center of the Circle is “+”, Always write positive motion in the equation first. g is still g in ΣF Equations Always = mv2/r

Sample Problem - Tension A ball on the end of a string is revolved at a uniform rate in a vertical circle of radius 75 cm. If its speed is 4.4 m/s and its mass is .35 kg, calculate the tension in the string when the ball is (a) at the top of its path, and (b) at the bottom of its path.

Sample Problem – Flat Curve A car of mass m is attempting to round an unbanked curve with a radius of r. If the coefficient of static friction between the tires and the road is ms, what is the maximum speed the driver can have and successfully negotiate the curve?

Other Ideas Normal Force Centrifugal Force?

Sample Problem – Banked Curve A car of mass m is attempting to round a curve with a banking angle of q, and a radius of r. If there is no friction on the road, what is the speed the driver can have to successfully negotiate the curve?

Rotational Mechanics Torque is the tendency of a force to produce rotation around an axis. Torque = Force x distance Distance is measured from the pivot point or fulcrum to the location of the force on the lever arm. The longer the lever arm, the greater the torque.

Balancing Torques The unit for torque is the newton-meter. When torques are balanced or in equilibrium, the sum of the torques = 0 t = (F1 x d1) + (F2 x d2) + ….. The Center of Gravity of an object is the point on an object that acts like the place at which all the weight is concentrated.

Rotational Inertia Rotational Inertia or moment of inertia is the resistance of an object to changes in its rotational motion. (rotating objects keep rotating, non-rotating objects tend to stay still) The further the mass is located from the axis of rotation, the greater the rotational inertia. Greater rotational inertia means more laziness per mass.

Rotational Inertia All objects of the same shape have the same laziness per mass. You can change your rotational inertia when spinning by extending your arms or legs. Angular momentum is the measure of how difficult it is to stop a rotating object. Angular momentum = mass x velocity x radius