1. Lecture 12: Elastic Potential Energy & Energy Conservation

2. Questions of Yesterday 1) A 50-kg student starting from rest slides down a frictionless waterslide of height 10 m while a 100-kg student slides down a similar slide that is only 5 m high. Which student is going faster when they reach the bottom? a) the 50-kg student b) the 100-kg student c) they are going the same speed 2) A women pulls a crate up a rough (with friction) inclined plane at a constant speed. Which statement is NOT true? a) The work done on the crate by the normal force of the inclined plane on the crate is ZERO b) The work done on the crate by gravity is ZERO c) The work done by the net force on the crate is ZERO d) The gravitational PE is increasing

3. Spring Potential Energy When you push or pull on a spring are you doing work on the spring? As you displace a spring from its equilibrium position (compress or stretch it) its potential to gain kinetic energy increases Does an object attached to a stretched or compressed spring have the potential to gain kinetic energy? Is mechanical energy conserved? Elastic Potential Energy energy stored in a spring that is compressed or stretched from its equilibrium position

4. Equilibrium Position Spring exerts no force x = 0 Stretching vs Compressing x = 0 -x +x x = -d x = 0 x = d x = 0 Compressed  x = negative F = positive +F -F Stretched  x = positive F = negative

5. Force Exerted by the spring ALWAYS acts in a direction OPPOSITE to displacement from equilibrium position Restoring Force x = -d x = 0 x = d x = 0 Compressed  x = negative F = positive +F -F Stretched  x = positive F = negative RESTORING FORCE of a spring always acts to restore spring to equilibrium (to return spring to x = 0)

6. Restoring Force (F S ) increases with displacement from equilibrium Restoring Force F S (N) x (m) x = 0 equilibrium x = d x = 0 stretched x = -dx = 0 compressed equilibrium stretched compressed k = spring constant (units = N/m) (measures stiffness of spring) x = displacement from eq. Position (x = 0) F S = -kx Spring Restoring Force

7. Just like gravitational potential energy…. Elastic Potential Energy = -(Work done by the spring) Work done by a Spring F S (N) x (m) equilibrium stretched compressed F S = -kx Work done by spring (W S ) is negative Work done by a varying force = Area under F vs x curve F S is not constant W S = F S x

8. Just like gravitational potential energy…. Elastic Potential Energy = -(Work done by the spring) Work done by a Spring FSFS x F S = -kx W S = (1/2)F S x Work done by a varying force = Area under F vs x curve A = (1/2)F S x W S = -(1/2)kx 2 PE S = (1/2)kx 2

9. Energy Conservation W nc =  KE +  PE  PE G  PE S when W nc = 0 (KE + PE G + PE S ) i = (KE + PE G + PE S ) f (1/2)mv 2 mgy (1/2)kx 2 W nc = (KE f - KE i ) + (PE Gf - PE Gi ) + (PE Sf - PE Si )

10. Practice Problem An archer pulls her bowstring back 0.400 m by exerting a force that increases uniformly from 0 to 250 N. What is the equivalent spring constant of the bow? How much work does the archer do in pulling the bow?

11. Practice Problem A toy gun uses a spring to project a 10.2 g soft rubber sphere horizontally. The barrel of the gun and the uncompressed spring are both 15 cm long, the spring constant is 10 N/m, and a constant frictional force of 0.05 N exists between the barrel and projectile. With what speed does the projectile leave the barrel if the spring was compressed 10.0 cm for this launch? If the gun is held 1.0 m above the ground what is the ball’s speed when it hits the ground?

12. Practice Problem A block of mass 10.0 kg slides from rest down a frictionless 30.0 o incline and is stopped by a spring with k = 100 N/m. The block slides 10.00 m from the point of release to the point where it comes to rest against the spring. When the block comes to rest, how far has the spring been compressed?

13. Questions of the Day 1) A mass with speed v hits a horizontal spring and compresses it a distance d. If the the speed of the mass were doubled (2v) what would the compression distance be? a) 4d b) 2d c) d d) d/2 2) A mass on a spring is oscillating back and forth from x = -d to x = d? At what point in the oscillation is the speed of the mass the greatest? a) x = d b) x = -d c) x = 0 d) x = d and x = -d