Elastic Potential Energy (Warning: Algebra required)
1. Consider a mass m attached to a horizontal spring of spring constant k resting on a frictionless table. All measurements are from the equilibrium rest point. An external applied force F is gradually applied to compress the spring until the force is just balanced by the spring force. Write a relation between F, k, and the maximum compression distance X. (Use a capital “X” for the maximum compression distance.)
2. Draw a Force-Displacement graph for this motion and from this calculate the work done on the spring. Write this elastic potential energy in terms of the maximum force and the compression distance.
3. The mass is then released. Because the system is frictionless, energy is conserved. Draw a diagram showing the types of energy in the system when the spring is at maximum compression, at its equilibrium position, and at maximum extension.
4. What is the elastic energy of the spring at these three positions? What does this mean for the maximum compression and maximum extension distances?
5. What is the kinetic energy of the mass at these three positions. From this, calculate the speed of the mass as it passes through the equilibrium position.
6. Write the total energy of the spring – mass system when the mass is at some arbitrary position x. Equate this to the total initial energy. From this, write a formula for the velocity of the mass at any position x. Because the term in the square root can’t be negative, what can you conclude about the sizes of X and x.
Answers: (1) F = k X (2) E = ½ Fmax X = ½ k X2 (4) E = ½ k X2, E = 0, E = ½ k X2 (5) 0, KE = ½ m v2, 0, v=Root(k/m) x (6) Energy = ½ k X2 = ½ k x2 + ½ m v2, v = ± ROOT(2/m [1/2 k { X2 – x2} ] )
Now consider a vertical spring of unextended length L. A mass m is attached to the spring and released. The mass oscillates up and down through a distance 2A (A stands for amplitude), and eventually comes to rest after a long time due to friction at distance A below the unextended length L. All distances x are measured from the unextended end of the spring.
7. What is the total energy of the spring-mass system just prior to release (elastic + kinetic + gravitational potential).
8. What is the total energy of the system when the mass comes to rest a distance 2A below the unextended length. Use this result to calculate the size of the amplitude of oscillation A in terms of k and m.
9. What is the total energy of the system when the mass is a distance A below the unextended length. Use this result to calculate the velocity at this point. What is the kinetic energy at this point. Relate this to the initial elastic energy – what do you notice.
10. After friction has acted for a long time, the mass comes to rest a distance A below the unextended length. Write the force balance equation to derive a relation for A in terms of m,g, and k. What is the total energy of the system now, noting that friction has removed the kinetic component.
Answers:(7)E = 0J (8) E = ½ k (2A)2 - m g 2A, A = m g / k (9) E = ½ k A2 + ½ m v2 – m g A, v = A Root(k/m), KE = ½ m v2 = ½ k A2(10) A = m g / k, E = -m g A + ½ k A2