1. A 5 kg block is attached to a horizontal unstretched spring of k = 350 N/m. The spring is stretched by 8 cm and set into oscillation.
→ Sketch the position vs. time graph and identify the 4 main positions - when the block is at the two amplitudes and both times the block crosses the equilibrium position
A. Find the frequency, period, and amplitude of the SHO.
B. Find the maximum acceleration of the block.
C. How much mechanical energy does the system have?
D. Find the maximum speed of the block.
E. Find the velocity and the acceleration of the block when it is at +1/2 * A
1. Do these answers need a sign? Why or why not?)
F. Find the position and velocity of the block at t = 4 seconds. What is the block’s momentum at that time? (these need a sign)
G. If the block has a position x = - 0.05 m, how much time has elapsed?
2. 3 kg block attached to a horizontal unstretched spring with a frequency of 1.5625 Hertz. The block has first has a position of -0.1 m at t = 0.4 seconds
A. Determine the shm’s period and spring constant.
B. What is the maximum displacement of the block from its equilibrium point?
C. What is the object’s maximum velocity and maximum acceleration?
D. What is the object’s positon, velocity, and acceleration at t = 3 seconds?
E. What is the elastic potential energy in the spring and what is the object’s kinetic energy at t = 2.5 seconds?
F. At what time is the object at position +0.1 m?
G. Draw the velocity vs time graph and the acceleration vs. time graph. Identify the four main positions on the graphs and put axis numbers on those.
Answers Below
Answers
A. T = 0.75 s, f = 1.33 Hz, A = 0.08 m
B. a = 5.6 m/s/s → no sign, because it is both the positive and negative max acceleration
C. 1.12 J
D. 0.67 m/s → no sign, because it is both the positive and negative max speed
E. 0.58 m/s; -2.8 m/s/s → no sign on velocity, because you don’t know whether the block is moving towards or away from equilibrium - you only know position; but you do know the acceleration is negative because acceleration and position always have the opposite sign
F. x = -0.04 m; v = -0.57 m/s; p = -2.9 kg*m/s → your calculator won't give you the negative value of the velocity because you are taking the square root, you have to figure it out using the picture. Velocity is negative because the block just completed a full cycle at 3.75 seconds, and would cross equilibrium at 3.9375 and get to other amplitude at 4.125, so it is moving toward negative amplitude after equilibrium, you can also just draw a position vs. time graph and see which direction the slope of the graph is (remember that velocity is the slope of the position vs. time graph)
G. t = 0.27 s
A. T = 0.64 s; k=289.15 N/m
B. A = 0.141 m
C. Vmax = 1.39 m/s; amax = 13.63 m/s/s
D. X = -0.054 m; a = -5.22 m/s/s; v = +1.28 m/s (last full cycle at 2.56 seconds, next equilibrium at 2.72 seconds, negative amplitude at 2.88 seconds, next equilibrium at 3.04 seconds --> moving from negative amplitude to equilibrium so positive velocity – also notice how close this velocity is to vmax, and the next equilibrium will be at 3.04 seconds)
E. Us = 1.44 J; ME = 2.89 J; K = 1.44 J
F. t = 0.08 s, 0.56 s, 0.72 s, 1.2 s, 1.36 s