Q23 How much work does it take to stretch a spring with k = 210 N/m
(a) 10 cm from equilibrium and
(b) from 10 cm to 20 cm from equilibrium?
(a) w = 1/2 k x^2 = 1/2 * 210 * .1^2 Joules
(b) 1/2 * 210 * .2^2 ... minus (a)
1.05
A23 Well, the amount of work required to stretch a spring actually depends on how hard you need to pull it. It's like trying to convince a lazy spring to get out of bed. So, without knowing the force applied, I'm afraid I can't help you calculate the exact amount of work. But hey, if you need someone to convince that lazy spring to get up, I can definitely lend a hand!
To calculate the work required to stretch a spring, we can use the formula:
Work = (1/2) * k * x^2
Where:
- Work is the amount of work done on the spring (in joules)
- k is the spring constant (in N/m)
- x is the displacement of the spring (in meters)
a) To stretch the spring 10 cm from equilibrium:
First, convert 10 cm to meters:
10 cm = 0.10 m
Plug the values into the formula:
Work = (1/2) * 210 N/m * (0.10 m)^2
Work = (1/2) * 210 N/m * 0.01 m^2
Work = (1/2) * 210 N/m * 0.01 m
Work = 1.05 joules
Therefore, it takes 1.05 joules of work to stretch the spring 10 cm from equilibrium.
b) To stretch the spring from 10 cm to 20 cm from equilibrium:
First, calculate the displacement:
Displacement = (20 cm - 10 cm) = 10 cm = 0.10 m
Plug the values into the formula:
Work = (1/2) * 210 N/m * (0.10 m)^2
Work = (1/2) * 210 N/m * 0.01 m^2
Work = (1/2) * 210 N/m * 0.01 m
Work = 1.05 joules
Therefore, it takes 1.05 joules of work to stretch the spring from 10 cm to 20 cm from equilibrium.
To calculate the work done in stretching a spring with a spring constant (k), we need to use Hooke's Law and the formula for work.
(a) To stretch the spring 10 cm from equilibrium:
1. Convert the distance to meters: 10 cm = 0.1 m
2. Apply Hooke's Law: F = k * x, where F is the force applied, k is the spring constant, and x is the displacement from equilibrium.
So, F = 210 N/m * 0.1 m = 21 N
3. Calculate the work done using the formula: work = (1/2) * k * x^2, where x is the displacement from equilibrium.
So, work = (1/2) * 210 N/m * (0.1 m)^2 = 1.05 J
Therefore, it takes 1.05 Joules of work to stretch the spring 10 cm from equilibrium.
(b) To stretch the spring from 10 cm to 20 cm from equilibrium:
1. Convert the distances to meters: 10 cm = 0.1 m, and 20 cm = 0.2 m
2. Apply Hooke's Law to find the force required at each displacement:
- For 10 cm:
F1 = k * x1 = 210 N/m * 0.1 m = 21 N
- For 20 cm:
F2 = k * x2 = 210 N/m * 0.2 m = 42 N
3. Calculate the work done for the whole displacement using the formula: work = (1/2) * k * (x2^2 -x1^2), where x1 and x2 are the initial and final displacements.
So, work = (1/2) * 210 N/m * ((0.2 m)^2 - (0.1 m)^2) = 4.2 J
Therefore, it takes 4.2 Joules of work to stretch the spring from 10 cm to 20 cm from equilibrium.