Recall: Uniform Circular Motion An object moving in a circle with constant speed, v, experiences a centripetal acceleration with: *a magnitude that is constant in time and is equal to *a direction that changes continuously in time and always points toward the center of the circular path For uniform circular motion, the velocity is tangential to the circle and perpendicular to the acceleration

Period and Frequency A circular motion is described in terms of the period T, which is the time for an object to complete one revolution. The distance traveled in one revolution is r The frequency, f, counts the number of revolutions per unit time.

Example of Uniform Circular Motion The moon’s nearly circular orbit about the earth has a radius of about 384,000 km and a period T of 27.3 days. Determine the acceleration of the Moon towards the Earth.

Moon…... *So we find that amoon / g = 0.000278 *Newton noticed that RE2 / R2 = 0.000273 *This inspired him to propose that Fgravity  1 / R2 (more on gravity in future lectures) amoon R RE g

Circular Motion When we see an object carrying out circular motion, we know that there must be force acting on the object, directed towards the center of the circle. When you look at the circular motion of a ball attached to a string, the force is provided by the tension in the string. When the force responsible for the circular motion disappears, e.g. by cutting the string, the motion will become linear.

Recall: Uniform Circular Motion *motion in a circle or circular arc at constant speed *the acceleration changes the direction of the velocity, not the magnitude *the “center-seeking” or centripetal acceleration is always orthogonal to the velocity and has magnitude: The period of the motion:

Or m 4π²r T² Centripetal Force Newton’s 2nd Law: The net force on a body is equal to the product of the mass of the body and the acceleration of the body. *The centripetal acceleration is caused by a centripetal force that is directed towards the center of the circle. Fc = mac Or m 4π²r T²

WARNING!!! Centripetal Force is not a new, separate force created by nature! Some other force creates centripetal force Swinging something from a string  tension Satellite in orbit  gravity Car going around curve  friction

Uniform Circular motion Fc = m (v2 / r) What is the physical meaning of the formula? What happens if the initial speed v of the object in uniform circular motion is forced to move faster? A greater T is needed, What happens if there is energy lost due to friction on the plane? v decreases  By Fc = mv2 / r, r decreases

Example A 1.25-kg toy airplane is attached to a string and swung in a circle with radius = 0.50 m. What was the centripetal force for a speed of 20 m/s? What provides the Fc? Fc = mv2/r = (1.25 kg)(20 m/s)2/(0.50m) = 1000 N Fc = 1000 N Tension in the string

Circular Motion and its Connection to Friction When you drive your car around a corner you carry out circular motion. In order to be able to carry out this type of motion, there must be a force present that provides the required acceleration towards the center of the circle. This required force is provided by the friction force between the tires and the road. But remember ….. The friction force has a maximum value, and there is a maximum speed with which you can make the turn. Required force = Mv2/r. If v increases, the friction force must increase and/or the radius must increase.

Questions: You are driving a car with constant speed around a horizontal circular track. On a piece of paper, draw a Free Body Diagram for the car. How many forces are acting on the car? A) 1 B) 2 C) 3 D) 4 E) 5 Normal Tension 3 forces, Tension (friction), gravity and the normal force Gravity The net force on the car is A. Zero B. Pointing radially inward C. Pointing radially outward B: Force of tension is pointing inward (centripetal force) 25

A car of mass, m, is traveling at a constant speed, v, along a flat, circular road of radius, R. Find the minimum µs required that will prevent the car from slipping

Question 3 Davian sits on the outer rim of a merry-go-round, and Diego sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds. Diego’s velocity is: Diego Davian (a) the same as Davian’s (b) Faster than Davian’s Slower than Davian’s a = V2/r (C) Diego is slower than the velocity of Davian