A spring has a relaxed length of 6 cm and a stiffness of 50 N/m. How much work must you do to change its length from 8 cm to 11 cm?
To find the work done to change the length of the spring, we need to calculate the work done in stretching the spring from its initial length to its final length.
The work done on a spring is given by the formula:
W = (1/2)k(x^2 - x0^2)
Where:
W = work done
k = stiffness of the spring
x = final length of the spring
x0 = initial length of the spring
Given:
k = 50 N/m
x0 = 6 cm
x = 11 cm
Converting the lengths to meters:
x0 = 6 cm = 0.06 m
x = 11 cm = 0.11 m
Substituting the values into the formula:
W = (1/2) * 50 N/m * (0.11^2 - 0.06^2)
Simplifying the exponents:
W = (1/2) * 50 N/m * (0.0121 - 0.0036)
Calculating:
W = (1/2) * 50 N/m * 0.0085
W = 0.425 N * m
Therefore, to change the length of the spring from 8 cm to 11 cm, you must do 0.425 N * m of work.
To find the amount of work required to change the length of a spring, we need to calculate the potential energy stored in the spring at the final and initial lengths, and then subtract the initial potential energy from the final potential energy. The potential energy stored in a spring can be calculated using the formula:
Potential energy = (1/2) * k * (delta x)^2
Where:
k is the stiffness of the spring (in N/m)
delta x is the change in length of the spring (in meters)
First, let's convert the lengths given in centimeters to meters:
Initial length = 8 cm = 0.08 m
Final length = 11 cm = 0.11 m
Now, let's calculate the potential energy at the initial and final lengths:
Initial potential energy = (1/2) * k * (delta x)^2
= (1/2) * 50 N/m * (0.08 m - 0.06 m)^2
Final potential energy = (1/2) * k * (delta x)^2
= (1/2) * 50 N/m * (0.11 m - 0.06 m)^2
Finally, subtract the initial potential energy from the final potential energy to find the amount of work done:
Work = Final potential energy - Initial potential energy
Simplify the equation and plug in the values to find the answer.
energy= 1/2 k x^2
change in stored energy= 1/2 (50)(1/.08^2-1/.11^2)