Work and Energy – Spring Compression Physics Problem

Problem 3: Spring Compression & Work-Energy Calculations

Question

A 1 kg block is dropped onto a relaxed vertical spring that has a spring constant k = 1000 N/m. The block compresses the spring by 10 cm.

While the spring is being compressed:

(a) What is the work done on the block by the gravitational force?
(b) What is the work done on the block by the spring force?
(c) What is the speed of the block just before it hits the spring?
(d) After being compressed, the spring is released. To what maximum height does the block rise?

Work done by gravity is calculated using:
W_g = mgd cos(θ)

Here, m = 1 kg, g = 9.8 m/s², d = 10 cm = 0.1 m, and angle θ = 0° (since force and displacement are in the same direction).

W_g = 1 × 9.8 × 0.1 = 0.98 J

(a) Work done by gravity = 0.98 J

Work done by a spring is given by:
W_s = -½ k x²

Where k = 1000 N/m, x = 10 cm = 0.1 m

W_s = -½ × 1000 × (0.1)² = -5 J

(b) Work done by the spring = -5 J

Using the work-energy theorem:
½ mv² = W_g – |W_s|

0.5 × 1 × v² = 5 – 0.98 = 4.02
v = √8.04 = 2.8 m/s

Using conservation of energy:
mgh = ½ kx²

Solve for h:
1 × 9.8 × h = 0.5 × 1000 × (0.1)² = 5
h = 5 / 9.8 = 0.51 m

(d) Maximum height = 0.51 m