A force of 250 N is required to stretch a spring 5m from rest. Using Hooke's law, F=kx, how much work, in joules, is required to stretch the spring 7m from rest?
The energy required scales with x^2.
Multiply 250 J by (7/5)^2.
350 J
To find the work required to stretch the spring 7m from rest, we can use Hooke's law and the equation for work.
Hooke's law states that the force required to stretch or compress a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be represented as:
F = k * x
Where:
F is the force applied to the spring (in Newtons)
k is the spring constant (in Newtons per meter)
x is the displacement from the equilibrium position (in meters)
Given that a force of 250 N is required to stretch the spring 5m from rest, we can plug these values into the equation and solve for k:
250 N = k * 5m
Simplifying the equation:
k = 250 N / 5m
k = 50 N/m
Now, we can use the equation for work to calculate the energy required to stretch the spring 7m from rest. The equation for work is given by:
Work = (1/2) * k * x^2
Substituting the values we already have:
Work = (1/2) * 50 N/m * (7m)^2
Work = (1/2) * 50 N/m * 49m^2
Simplifying the equation:
Work = 24.5 N/m * 49m^2
Work = 24.5 N/m * 2401m^2
Work = 58,789.5 J (joules)
Therefore, it requires 58,789.5 joules of work to stretch the spring 7m from rest.
To find the amount of work required to stretch the spring 7m from rest using Hooke's law, we need to use the relationship between force, displacement, and the spring constant.
Hooke's law states that the force required to stretch or compress a spring is directly proportional to the displacement. The formula is F = kx, where F is the force, k is the spring constant, and x is the displacement.
In this case, we are given the force required to stretch the spring 5m from rest, which is 250 N. We can use this information to find the spring constant, k. Rearranging the formula, we get k = F / x = 250 N / 5 m = 50 N/m.
Now that we know the spring constant, we can find the amount of work required to stretch the spring 7m from rest. The displacement has increased from 5m to 7m, so we need to calculate the energy required for this additional 2m of displacement.
Using the fact that the energy required scales with x^2, we can use the following formula:
Work = (Force * (displacement / reference displacement)^2)
Substituting the given values, we get:
Work = (250 J * (7m / 5m)^2)
Simplifying this equation, we have:
Work = (250 J * (49/25))
Work = 490 J
Therefore, the amount of work required to stretch the spring 7m from rest is 490 joules.