## Work Done by the Spring Forces

Consider a spring with spring constant k and force-free length Lo. As shown in Fig. 8.5, the spring is force free at time to. At later times it is at first compressed (t1) and subsequently stretched (t2). Let Xj denote the displacement of the end of the spring marked with symbol A along the direction of the unit vector e at time tj. Then the force exerted by the spring on the mass m at time tj can be written as

Note that, for the spring shown in Fig. 8.5, Xi > 0 at time t1 and thus the spring exerts a force on mass m in the - e direction. On the other hand, the spring force is in the direction of e at time t2 because x2 < 0.

Figure 8.5. The force exerted by a spring on a mass m that is in contact with the spring at point A. The spring is force free at time to, neither stretched nor compressed. The spring is compressed at time i1, and the force exerted by the spring is in the opposite direction of the velocity of the point A. Therefore, in this configuration, the power exerted by the spring on the mass m is negative, indicating that it causes a reduction in the kinetic energy of mass m. The spring is under tension at time t2, and the power produced by the spring is again negative.

The power exerted by the equal to the scalar product of the spring force and velocity of the point of application of the force:

P = (-kx) (dx/dt) => W = -(V2) kx22 + (V2) kxx2 (8.19b)

If both ends of a spring undergo displacements, the work done by the spring on the objects at both ends of the spring is given by the equation:

in which 82 and S1 represent the extension (compression) of the spring at times t2 and t1, respectively. The work done by the spring force is independent of the path taken during compression or tension, and thus the spring force is conservative. If we call the term (1/2) k82 the potential energy Vs of the spring, then we have

in which Vs2 and Vs1 denote the potential energy of the spring at times t2 and t1, respectively.

## Getting Started With Dumbbells

The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.

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