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3. Centripetal Forces Rotational Motion and Astrophysics Advanced Higher Recap Angular Equations of Motion 0 t 1 2 0 t t 2 Motion in a Circle s r v r d dt at r d d 2 dt dt 2 0 2 2 2 Central (Radial) Acceleration Consider a particle undergoing circular motion The particle travels from A to B in time Δt and with speed v. The change in speed is given as Δv arcAB 2r t v v Average Acceleration: v 2v sin 2v sin v 2 sin aav 2 r t t r v When Θ is small and is measured in radians this means that sin Θ = Θ v v a a r r 2 2 Since v r r 2 a a r r 2 2 Centripetal Force The centripetal force is the one felt when you move round in a circle. This is due to you accelerating into the centre of a circle. You feel like you are being pulled outwards…but you are in fact still moving in a straight line. Magnitude of the Force The same rule still applies for working out the force being applied to an accelerating object F ma 2 and v 2 a r r Therefore 2 mv 2 F mr r Car on a banked track Same rules of “box on a slope” still apply R is the “normal reaction force of the slope, and Fr is the frictional force of the slope. We must find the horizontal and vertical component of each force. For this example we assume that friction is reduced to zero. Vertically (Equation 1) R cos mg This motion has zero acceleration Horizontally (Equation 2) mv 2 R sin r This motion has constant central acceleration Divide Equation 2 by Equation 1 to get: 2 v tan gr Vertical Circular Motion Similar ideas have to be considered when discussing vertical, but circular motion Highest Point mv Ttop mg r Lowest Point mv 2 Tbottom mg r 2 2 Ttop mv mg r Tbottom Thoriz mv r mv 2 mg r 2 As there is no vertical component of tension at this point Example 1. A piece of string has a breaking force of 56N. This string is used to whirl a mass of 150g in a horizontal circle. (a)The 150g mass moves in a horizontal circle of radius 1.2m. Calculate the maximum angular velocity of the mass. (b)The mass is now rotated at 85rpm. Find the maximum possible radius of the circle. Solution (a) F mr 2 (b) 56 150 x10 1.2 3 56 3 150 x10 1.2 17.6 rads -1 2 85 2 8.9 rads -1 85rpm 60 F mr 2 56 150 x10 3 r 8.9 56 4.7m r 3 150 x10 8.9 8.9 2