Spring Force and Hooke’s Law We next want to examine the work done on a particle-like object by a particular type of variable force—namely, a spring force, the force from a spring. Many forces in nature have the same mathematical form as the spring force. Thus, by examining this one force, you can gain an understanding of many others. The spring force is given by Hooke’s law, where k is the spring constant.

7-7 The Work Done by a Spring Force

The Work Done by an Applied Force Now suppose that we displace the block along the x axis while continuing to apply a force to it. During the displacement, our applied force does work, Wa on the block while the spring force does work, Ws. The change in the kinetic energy of the block due to these two energy transfers is: If the block is stationary before and after the displacement, then Kf and Ki are both zero and the above equation reduces to:

Sample Problem A cumin canister of mass, m = 0.40 kg slides across a horizontal frictionless counter with speed, v= 0.50 m/s. It then runs into and compresses a spring of spring constant, k = 750 N/m. When the canister is momentarily stopped by the spring, by what distance d is the spring compressed?

7.8 Work Done by a General Variable Force One-Dimension: Three-Dimension: P73, p165: A force in the positive direction of an x axis acts on an object moving along the axis. If the magnitude of the force is F = 10e-x/2.0 N, with x in meters, find the work done by as the object moves from x = 0 to x = 2.0 m by (a) plotting F(x) and estimating the area under the curve and (b) integrating to find the work analytically.

7.9   Power The time rate at which work is done by a force is said to be the power due to the force. If a force does an amount of work W in an amount of time Δt, the average power due to the force during that time interval is, The instantaneous power P is the instantaneous time rate of doing work, which we can write as, The SI unit of power: joule per second = watt (W). P46,p163: The loaded cab of an elevator has a mass of 3.0 × 103 kg and moves 210 m up the shaft in 23 s at constant speed. At what average rate does the force from the cable do work on the cab?