a spring with a spring constant 30.0 N/m is stretched 0.200 m from its equilibrium position. How much work must be done to stretch it an additional 0.100 m?
It equals the potential energy change.
(1/2)k [(0.3)^2 - (0.2)^2] (Joules)
To calculate the work done to stretch the spring an additional 0.100 m, we need to use the equation for the potential energy stored in a spring:
Potential Energy (PE) = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
Given:
Spring constant (k) = 30.0 N/m
Initial displacement (x) = 0.200 m
Additional displacement (Δx) = 0.100 m
First, let's calculate the potential energy at the initial displacement:
PE_initial = (1/2) * k * x^2
Substituting the values into the equation:
PE_initial = (1/2) * 30.0 N/m * (0.200 m)^2
PE_initial = 0.6 J
Next, let's calculate the potential energy at the final displacement:
PE_final = (1/2) * k * (x + Δx)^2
Substituting the values into the equation:
PE_final = (1/2) * 30.0 N/m * (0.200 m + 0.100 m)^2
PE_final = 0.9 J
The work done is the change in potential energy. Therefore, the work done to stretch the spring an additional 0.100 m is:
Work done = PE_final - PE_initial
Work done = 0.9 J - 0.6 J
Work done = 0.3 J
So, 0.3 Joules of work must be done to stretch the spring an additional 0.100 m.