Which of the five graphs correctly shows the potential energy of a spring as a f

Question

Which of the five graphs correctly shows the potential energy of a spring as a function of its elongation x? I II III IV V A 0.50-kg block attached to an ideal spring with a spring constant of 80 N/m oscillates on a horizontal frictionless surface. When the spring is 4.0 cm longer than its equilibrium length, the speed of the block is 0.50 m/s. The greatest speed of the block is: 0.23 m/s 0.32 m/s 0.55 m/s 0.78 m/s 0.93 m/s 13. The string in the figure is 50 cm long. When the ball is released from rest, it swings along the dotted are. How fast is it going at the lowest point in its swing? 2.0 m/s 2.2 m/s 3.1 m/s 4.4 m/s 6.0 m/s

Explanation / Answer

11) ANS : C

12)

Spring Energy = 1/2KX^2

Kinetic Energy = 1/2mv^2

At the time given, our initial time for our initial energy, it has some kinetic energy and some spring energy.
At the time of it's greatest speed, our final time or final energy, it will be at equilibrium, x=0 and will have all kinetic energy. Since Energy Initial = Energy Final we can set spring energy + kinetic energy at 4 cm = kinetic energy at 0 cm.
Therefore,

1/2(80)(0.04)^2 + 1/2(0.5)(0.5)^2 = 1/2(0.5)(v)^2

0.064 + 0.0625 = 0.25v^2

v^2 = 0.506

v = 0.71 m/s

13)

Taking Ugrav= 0 at the bottom of the swing,

at the top E = K +U = 0 + mg x 0.50

At the bottom Ef = Kf + Uf = mv + 0.

Equating these two gives

v = 2g x 0.50

v = sqrt(2 x 9.8 x 0.50)

v = 3.1m/s

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