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Unformatted text preview: Sample Problems (contd.) The difficulty of each problem is listed immediately after the problem number. Problem 1 (medium): A block of mass m is attached to a spring with spring constant k , and undergoes undamped vertical oscillations with amplitude A . Let y = 0 denote gravitational equilibrium, and let upward denote the positive direction, and downward the negative direction. 1: How long does the block take to move from equilibrium at y = 0 upward to y = A/ 2? 2: How long does the block take to move from y = A/ 2 upward to y = A ? 3: What minimum amount of time does the block take to go from y =- A √ 3 / 2, moving downward, to y = A/ √ 2, moving upward? 4: At a time t after the block has passed equilibrium, moving in the upward direction, what restoring force is exerted by the spring? At this time, what is the position of the block relative to the spring’s unstretched length (before the mass was attached)? Solution: The angular frequency of the simple harmonic motion is ω = r k m We first consider part 1 . We assume here that at time t = 0 the block starts out at y = 0. The time t 1 at which the block reaches y = A/ 2 is then given by A 2 = A sin( ωt 1 ) Solving for t 1 , we find t 1 = 1 ω arcsin 1 2 ¶ = π 6 ω = π 6 r m k We now consider part 2 . The time taken to go from y = 0 to y = A is a quarter-period. Therefore, the time taken to go from y = A/ 2 to y = A is a quarter-period minus the time we calculated in part 1: t 2 = T 4- t 1 = π 2 r m k- π 6 r m k = π 3 r m k Notice that in parts 1 and 2, the block is traveling the same distance. But the time required is longer in part 2 than in part 1. This is because in oscillatory motion, the object travels more slowly the further it is from equilibrium. 1 We now consider part 3 . We break this into two parts. Let’s first find the time t 3 a taken to go from y =- A √ 3 / 2 down to y =- A . Then we find the time t 3 b it takes for the block to go from y =- A up to y = A/ √ 2. The desired time is the sum of t 3 a and t 3 b...
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- Spring '09
- Simple Harmonic Motion, Sin, Fspr