Question

A 100-g block slides back and forth on a frictionless surface between two springs, as shown in Fig. 7.18. The left-hand spring has k = 110 N/m and its maximum compression is 21 cm.The right-hand spring has
k = 240 N/m. Find (a) the maximum compression of the right-hand spring and (b) the speed of the block as it moves between the springs.

Answer 1

(a) Energy is conserved at every point in the block's motion, so the potential energy P stored in the first spring at its maximum compression is the same as is stored in the second spring.

The total work performed on the block by the first spring is

W = -1/2 (110 N/m) (0.21 m²) = -2.4255 J

The work performed by the second spring is the same, so

W = -1/2 (240 N/m) x²

Solve for x :

x² = -2W/(240 N/m) = 0.0202125 m²

x ≈ 0.14 m = 14 cm

(b) By the work-energy theorem, the total work performed by either spring on the block as the spring is compressed is equal to the change in the block's kinetic energy. The restoring force of the spring is the only force involved. At maximum compression, the block has zero velocity, while its kinetic energy and hence speed is maximum just as it comes into contact with either spring.

W = 0 - K

W = -1/2 (0.10 kg) v²

v² = -2W/(0.10 kg) = 48.51 m²/s²

v ≈ 7.0 m/s