Tom has built a large slingshot, but it is not working quite right. He thinks he can model the slingshot like an ideal spring with a spring constant of 75.0 N/m. When he pulls the slingshot back 0.375 m from a nonstretched position, it just does not launch its payload as far as he wants. His physics professor "helps" by telling him to aim for an elastic potential energy of 22.0 J. Tom decides he just needs elastic bands with a higher spring constant. By what factor does Tom need to increase the spring constant to hit his potential energy goal? factor of spring constant increase: During a follow-up conversation, Tom's physics professor suggests that he should leave the slingshot alone and try pulling the slingshot back further without changing the spring constant. How many times further than before must Tom pull the slingshot back to hit the potential energy goal with the original spring constant? factor of pullback distance increase: In which of the two scenarios does Tom have to pull harder? they are equal increased pullback distance increased spring constant

A horizontal spring attached to a wall has a force constant of k = 850 N/m. A block of mass m = 1.00 kg is attached to the spring and rests on a frictionless, horizontal surface as in Figure P8.35. (a) The block is pulled to a position xi = 6.00 cm from equilibrium and released. Find the elastic potential energy stored in the spring when the block is 6.00 cm from equilibrium and when the block passes through equilibrium. (b) Find the speed of the block as it passes through the equilibrium point. (c) What is the speed of the block when it is at a position xi/2 = 3.00 cm? (d) Why isnt the answer to part (c) half the answer to part (b)? Figure P8.35

A block is connected to a spring that is suspended from the ceiling. Assuming air resistance is ignored, describe the energy transformations that occur within the system consisting of the block, the Earth, and the spring when the block is set into vertical motion.

A small object is attached to two springs of the same length l, but with different spring constants k1 and k2 as shown in Figure P9.31. Initially, both springs are relaxed. The object is then displaced straight along the x axis from xi to xf. Find an expression for the work done by the springs on the object.

As a simple pendulum swings back and forth, the forces acting on the suspended object are the force of gravity, the tension in the supporting cord, and air resistance, (a) Which of these forces, if any, does no work on the pendulum? (b) Which of these forces does negative work at all times during the pendulums motion? (c) Describe the work done by the force of gravity while the pendulum is swinging.

A block of mass m = 5.00 kg is released from point and slides on the frictionless track shown in Figure P8.3. Determine (a) the blocks speed at points and and (b) the net work done by the gravitational force on the block as it moves from point to point . Figure P8.3

Rubber tends to be nonlinear as an elastic material. Suppose a particular rubber band exerts a restoring force given by Fx(x) = Ax Bx2, where the empirical constants are A = 14 N/m and B = 3.3 N/m2 so that Fx is in newtons when x is in meters. Calculate the change in elastic potential energy of the rubber band when an external force stretches it from x = 0 to x = 0.20 m.

For the potential energy curve shown in Figure P7.38, (a) determine whether the force Fx is positive, negative, or zero at the five points indicated. (b) Indicate points of stable, unstable, and neutral equilibrium. (c) Sketch the curve for Fx versus x from x = 0 to x = 9.5 m. Figure P7.38

A 60.0-kg athlete leaps straight up into the air from a trampoline with an initial speed of 9.0 m/s. The goal of this problem is to find the maximum height she attains and her speed at half maximum height. (a) What are the interacting objects and how do they interact? (b) Select the height at which the athletes speed is 9.0 m/s as y = 0. What is her kinetic energy at this point? What is the gravitational potential energy associated with the athlete? (c) What is her kinetic energy at maximum height? What is the gravitational potential energy associated with the athlete? (d) Write a general equation for energy conservation in this case and solve for the maximum height. Substitute and obtain a numerical answer. (e) Write the general equation for energy conservation and solve for the velocity at half the maximum height. Substitute and obtain a numerical answer.

A block is attached to a spring, and the block makes contact with a frictionless surface. Sketch a graph of the potential energy of the blockspring system as a function of position along with the corresponding system as in Figure 8.16 (page 224). Use this sketch to show how the block can move to the left from a positive position to a negative position and decrease the elastic potential energy of the system.