- Slides: 10
Aim: How do we characterize elastic potential energy?
Hooke’s Law states that the force required to stretch a spring is equal to the product of the spring’s spring constant and the amount it is being stretched or compressed. F = - k ∆x F = force, k = spring constant ∆x = displacement from equil.
Elastic Potential Energy We can show that the PE that a spring has due to being compressed or stretched can be expressed using, Us = ½ k (∆x)2 Us = elastic PE, k = spring constant, ∆x = change in length from equilibrium
Elastic Potential Energy Problems Ex. 1 A spring has a spring constant, k, of 320 N/m. How much must the spring be compressed to store 50 J? Us=1/2 kx 2 50=1/2(320)x 2 x=0. 56 m Ex. 2 A force of 10 N stretches a spring by 0. 2 m. What is the spring constant of the spring? F=-kx 10=k(. 2) k=50 N/m
More Elastic Potential Energy Problems Ex. 3 How far must you stretch a spring with k=10 N/m to store 200 J of energy? Us =1/2 kx 2 200=1/2(10)x 2 x=2 m Ex 4 When a 13. 2 -kg mass is placed on top of a vertical spring, the spring compresses 5. 93 cm. Find the force constant of the spring. F=-kx 132=k(0. 0593) F=mg=13. 2(10)=132 k=2, 225. 97 N/m and x = 0. 0593
More Elastic Potential Energy Problems Ex 5 A stretched spring stores 2. 0 J of energy. How much energy will be stored if the spring is stretched three times as far? Us=1/2 kx 2 So tripling the displacement, will multiply the potential energy by a factor if 9 so the spring stores 18 J of energy
Oscillating Spring Thought Questions The spring is stretched and the mass is released from position (a) and oscillates through to position (e) At what point(s) does the mass have the greatest potential energy? A, C, E At what point(s) does the mass have the greatest kinetic energy? B, D At what point(s) does the mass have the greatest acceleration? A, C, E At what point(s) does the mass have the smallest acceleration? B, D
Elastic Potential Energy and Kinetic Energy 6. A student places her 500 g physics book on a frictionless table. She pushes the book against a spring, compressing the spring by 4. 0 cm, then releases the book. What is the book’s speed as it slides away? The spring constant is 1250 N/m. 1/2 kx 2=1/2 mv 2 x=4 cm=0. 04 m kg ½(1250)(0. 04)2=1/2(0. 5)v 2 mass=500 g =0. 5
Elastic Potential and Kinetic Energy 7. A block sliding along a horizontal frictionless surface with speed v collides with a spring and compresses it by 2. 0 cm. What will be the compression if the same block collides with the spring at a speed of 2 v? 1/2 mv =1/2 kx 2 2 So we see that v and x are linearly related. Thus doubling v will double x so the compression is 4 cm
Elastic Potential Energy and Kinetic Energy 8. As a 15000 kg jet plane lands on an aircraft carrier, its tail hook snags a cable to slow it down. The cable is attached to a spring with spring constant 60, 000 N/m. If the spring stretches 30 m to stop the plane, what was the plane’s landing speed? ½mv 2=1/2 kx 2 ½(15000)v 2= ½(60000)(30)2 v=60 m/s