It takes 9.10 J of work to stretch a spring 5.10 cm from its unstressed length. What is the spring constant k in N/m?
To find the spring constant (k) given the work done (W) and the displacement (x), you can use Hooke's Law. Hooke's Law states that the force (F) exerted by a spring is directly proportional to its displacement. The formula is:
F = kx
Where F is the force, k is the spring constant, and x is the displacement.
In this case, you know the work done (W) on the spring, which is equal to the energy stored in the spring when it is stretched. The work done can be calculated using the formula:
W = (1/2)kx²
Now, you have two equations: F = kx and W = (1/2)kx². Since both equations have k and x, you can solve for k by using substitution.
Step 1: Rearrange the equation F = kx to solve for k:
k = F / x
Step 2: Substitute k in the equation W = (1/2)kx² with the expression obtained from the previous step:
W = (1/2)(F / x)x²
Step 3: Simplify the equation by canceling out the x terms:
W = (1/2)Fx
Step 4: Rearrange the equation to solve for k:
k = 2W / x²
Now you can plug in the given values:
W = 9.10 J
x = 5.10 cm = 0.0510 m
k = 2(9.10 J) / (0.0510 m)²
Solving this equation will give you the spring constant, k, in N/m.