Chapter 3 Circular Motion Uniform circular motion is the motion of an object in a circle with a constant or uniform speed. Speed When moving in a circle, an object traverses a distance around the perimeter of the circle. where R represents the radius of the circle and T represents the period. Velocity Velocity is a vector quantity. Velocity, being a vector, has both a magnitude and a direction. The magnitude of the velocity vector is the instantaneous speed of the object. The direction of the velocity vector is directed in the same direction that the object moves. Since an object is moving in a circle, its direction is continuously changing. So while the magnitude of the velocity vector may be constant, the direction of the velocity vector is changing. The best word that can be used to describe the direction of the velocity vector is the word tangential. The direction of the velocity vector at any instant is in the direction of a tangent line drawn to the circle at the object's location.
Acceleration An object travelling at a constant speed is undergoing acceleration if the direction of the velocity is changing. This is certainly true for uniform circular motion. The type of acceleration that occurs in uniform circular motion is called centripetal acceleration. Centripetal acceleration is an instantaneous acceleration. The Direction of Centripetal Acceleration So the direction of the acceleration is towards the center of the circle and centripetal acceleration is also known as radial acceleration (directed along the radius) The Magnitude of Centripetal Acceleration The equation for calculating centripetal acceleration is as follows
Let's look at where this equation comes from: Consider an object going around a circle, of radius r, with a velocity, v. If the object had continued travelling in a straight line as opposed to being forced into a circular path, then the displacement would be: d = v t. Because the object stayed on the circle, the displacement was not equal to d. This difference in displacement that would have been covered versus what is covered can be referred to as x, and this is the displacement that occurred due to the acceleration. This difference in displacement is due to the centripetal acceleration is the second part in d = v i t+ ½ at 2 A right triangle is formed by the vectors drawn. Applying the Pythagorean Theorem: r 2 + (vt) 2 = (r + x) 2 r 2 + v 2 t 2 = r 2 + 2rx + x 2 v 2 t 2 = 2rx + x 2 sub x = ( ½ at 2 ) v 2 t 2 = 2r( ½ at 2 ) + ( ½ at 2 ) 2 v 2 t 2 = rat 2 + ¼ a 2 t 4 As t becomes very small, ¼ a 2 t 4 becomes negligible v 2 t 2 = rat 2 a c = v 2 /r This makes sense because as the speed of an object in circular motion increases at a constant radius, the direction of the velocity changes more quickly requiring a larger acceleration; as the radius becomes larger (at a constant speed), the direction changes more slowly, meaning a smaller acceleration.
There are two other variations of the equation that are useful, depending on the information available in a situation. Since v =!!"! a = 2πr T r! a = 4π! r T! Recall also that T = 1/f So a = 4π! rf! Notice that in the equations above, the arrows above the vector symbols have been omitted, even though we are talking about vector quantities. As has been mentioned before, this means the magnitude of the vectors is being considered in the equations, not the direction. This is common convention in uniform circular motion questions since: the direction of centripetal acceleration is always toward the centre of the circle, the direction of centripetal force is always toward the centre of the circle, and the direction of the instantaneous velocity is always tangent to the circle. Ex A child on a merry- go- round is 4.4 m from the centre of the ride, travelling at a constant speed of 1.8 m/s. Determine the magnitude of the child s centripetal acceleration.
Find the magnitude and direction of the centripetal acceleration of a piece of lettuce on the inside of a rotating salad spinner. The spinner has a diameter of 19.4 cm and is rotating at 780 rpm (revolutions per minute). The rotation is clockwise as viewed from above. At the instant of inspection, the lettuce is moving eastward. Homework P 122 # 1,2 P 123 #3,4 P 126 # 5,6,8,9, P 127 # 5,6