Spring Oscillation Problems

1. Problmes-1

2. Problem 1At what point during the oscillation of a spring is the force on the mass greatest? Solution: Recall that F = - kx . Thus the force on the mass will be greatest when the displacement of the block is maximum, or when x = ±xm . • Problem 2 : What is the period of oscillation of a mass of 40 kg on a spring with constant k = 10 N/m? Sol: • Notice that period, frequency and angular frequency are properties of the system, not of the conditions placed on the system. FCI

3. Problem 3: A mass of 2 kg is attached to a spring with constant 18 N/m. It is then displaced to the point x = 2 . How much time does it take for the block to travel to the point x = 1 ? • Solution: For this problem we use the sin and cosine equations we derived for simple harmonic motion. Recall that x = xm cos(wt) . We are given x and xm in the question, and must calculate w before we can find t . x = xm cos(wt), 1=2 cos(3t),,,, t = 0.35 sec FCI

4. Problem 4: A mass of 2 kg oscillating on a spring with constant 4 N/m passes through its equilibrium point with a velocity of 8 m/s. What is the energy of the system at this point? From your answer derive the maximum displacement, xm of the mass. Since this is the total energy of the system, we can use this answer to calculate the maximum displacement of the mass. When the block is maximally displaced, it is at rest and all of the energy of the system is stored as potential energy in the spring, given by U = 1/2 kxm2 , Since energy is conserved in the system, we can relate the answer we got for the energy at one position with the energy at another: FCI

5. Since energy is conserved in the system, we can relate the answer we got for the energy at one position with the energy at another: • 5- A 200-g block is attached to a horizontal spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is 2.00 J, find (a) the force constant of the spring and (b) the amplitude of the motion. • 6- A block–spring system oscillates with an amplitude of 3.50 cm. If the spring constant is 250 N/m and the mass of the block is 0.500 kg, determine (a) the mechanical energy of the system, (b) the maximum speed of the block, and (c) the maximum acceleration. FCI

6. 7- A block of unknown mass is attached to a spring with a spring constant of 6.50 N/m and undergoes simple harmonic motion with an amplitude of 10.0 cm. When the block is halfway between its equilibrium position and the end point, its speed is measured to be 30.0 cm/s. Calculate (a) the mass of the block, (b) the period of the motion, and (c) the maximum acceleration of the block. FCI

7. 8 - A 200-g block is attached to a horizontal spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is 2.00 J, find (a) the force constant of the spring and (b) the amplitude of the motion. • 9- A particle executes simple harmonic motion with an amplitude of 3.00 cm. At what position does its speed equal half its maximum speed? FCI

8. 10 - A man enters a tall tower, needing to know its height. He notes that a long pendulum extends from the ceiling almost to the floor and that its period is 12.0 s. • How tall is the tower? • (b) What If ? If this pendulum is taken to the Moon, where the free-fall acceleration is 1.67 m/s2, what is its period there? FCI