Step-by-step explanation:
(a) The graph of Force vs. Stretch for the given spring can be represented by a straight line passing through the origin with a slope equal to the spring constant. The equation of the line is:
Force = Spring constant × Stretch
F = kx
where k = 200 N/m and x is the stretch of the spring in meters. The graph is shown below:
Force vs. Stretch graph for a spring with k = 200 N/m
(b) When a mass of 0.8 kg hangs from the spring, it experiences a force due to gravity equal to:
F = m × g = 0.8 kg × 10 m/s² = 8 N
Since the spring is in equilibrium, the force exerted by the spring must be equal and opposite to the force due to gravity. Therefore, the stretch of the spring is given by:
F = kx
x = F/k = 8 N / 200 N/m = 0.04 m
The point corresponding to this stretch is marked on the graph as shown below:
Force vs. Stretch graph with a point for a hanging mass of 0.8 kg
(c) The potential energy stored in the spring when it is stretched from zero to 0.06 meters can be calculated using the formula:
U = (1/2) k x²
U = (1/2) × 200 N/m × (0.06 m)² = 0.36 J
(d) The work done to stretch the spring from 0.1 meters to 0.16 meters can be calculated by finding the area under the Force vs. Stretch graph between these two stretches. This represents the change in potential energy of the spring due to the stretching. The work done is given by:
W = ΔU = U₂ - U₁
where U₁ and U₂ are the potential energies of the spring at stretches of 0.1 m and 0.16 m, respectively.
Using the formula for potential energy, we have:
U₁ = (1/2) k x₁² = (1/2) × 200 N/m × (0.1 m)² = 1 J
U₂ = (1/2) k x₂² = (1/2) × 200 N/m × (0.16 m)² = 2.56 J
Therefore, the work done is:
W = ΔU = U₂ - U₁ = 2.56 J - 1 J = 1.56 J
The area under the graph representing this work is shown below:
Force vs. Stretch graph with shaded area representing work done