A spring with spring constant of 30 N/m is stretched 0.19 m from its equilibrium position. How much work must be done to stretch it
an additional 0.077 m? Answer in units of J
Using the equation for spring potential energy
W=(1/2)kx²
so
ΔW=(1/2)k(x2²-x1²)
=(1/2)(30 N/m)*(0.96²-0.19²)
=13.3 J
Well, the work done on the spring can be calculated using the formula:
Work = (1/2) * k * x^2
Where k is the spring constant and x is the displacement from the equilibrium position.
So, let's calculate the work required to stretch the spring an additional 0.077 m:
Work = (1/2) * 30 N/m * (0.077 m)^2
Work = 0.5 * 30 N/m * 0.005929 m^2
Work = 0.08893 J (rounded to five decimal places)
So, the work required to stretch the spring an additional 0.077 m is approximately 0.08893 J.
To find the work required to stretch the spring an additional distance, we can use the formula for the potential energy stored in a spring:
Potential energy (U) = (1/2) * k * x^2
Where:
k = spring constant (given as 30 N/m)
x = distance stretched from the equilibrium position
First, let's find the initial potential energy of the spring when it is stretched 0.19 m:
U1 = (1/2) * 30 * (0.19)^2
U1 = 1.71 J
Next, we will find the potential energy when it is stretched an additional 0.077 m:
U2 = (1/2) * 30 * (0.19 + 0.077)^2
U2 = (1/2) * 30 * (0.267)^2
U2 = 0.642 J
Finally, to find the work done, we subtract the initial potential energy from the final potential energy:
Work (W) = U2 - U1
W = 0.642 - 1.71
W = -1.07 J (negative sign indicates work is done on the spring)
Therefore, the work required to stretch the spring an additional 0.077 m is approximately -1.07 J.
To find the work done to stretch the spring an additional distance, we need to use the formula for the potential energy stored in a spring:
Potential Energy (PE) = (1/2) * k * x^2
Where:
- PE is the potential energy stored in the spring
- k is the spring constant (in this case, 30 N/m)
- x is the displacement from the equilibrium position
The work done to stretch the spring is equal to the change in potential energy. Let's calculate it step by step:
1. Calculate the initial potential energy (PE1) for the spring when it is stretched 0.19 m:
PE1 = (1/2) * k * x^2
= (1/2) * 30 N/m * (0.19 m)^2
≈ 0.54 J (rounded to two decimal places)
2. Calculate the final potential energy (PE2) for the spring when it is stretched an additional 0.077 m:
PE2 = (1/2) * k * x^2
= (1/2) * 30 N/m * (0.19 m + 0.077 m)^2
= (1/2) * 30 N/m * (0.267 m)^2
≈ 0.86 J (rounded to two decimal places)
3. Determine the work done (W) to stretch the spring the additional distance:
W = PE2 - PE1
= 0.86 J - 0.54 J
≈ 0.32 J (rounded to two decimal places)
Therefore, to stretch the spring an additional 0.077 m, about 0.32 J of work must be done.