A spring has a natural length of 24 cm. If a 27-N force is required to keep it stretched to a length of 30 cm, how much work W is required to stretch it from 24 cm to 27 cm? (Round your answer to two decimal places.)
Well, well, well! Looks like we've got ourselves a springy question here. Let me stretch my clown brain a bit and see what comes out.
First, let's find the work done to stretch the spring from 24 cm to 30 cm. Since we know that a force of 27 N is required to keep it stretched to 30 cm, we can use the formula:
Work = Force x Distance
Work = 27 N x (30 cm - 24 cm)
Work = 27 N x 6 cm
Work = 162 N·cm
Now, for the grand finale! To find the work required to stretch the spring from 24 cm to 27 cm, we can use the same formula:
Work = Force x Distance
But here's the twist! The force required to keep the spring stretched to 27 cm isn't given. So we need to figure that out first. Since we know that it takes 27 N to keep it stretched to 30 cm, and the natural length of the spring is 24 cm, we can use proportions to find the force required for 27 cm.
27 cm is to 30 cm as X (unknown force) is to 24 cm.
X = (27 cm / 30 cm) * 27 N
X ≈ 24.3 N
Now that we have the force, we can find the work:
Work = 24.3 N x (27 cm - 24 cm)
Work = 24.3 N x 3 cm
Work = 72.9 N·cm
And there you go! The work required to stretch the spring from 24 cm to 27 cm is approximately 72.9 N·cm. Voila!
To find the work required to stretch the spring from 24 cm to 27 cm, we need to calculate the change in potential energy.
The potential energy of a stretched or compressed spring is given by the equation:
P.E. = 0.5 * k * (x - L)^2
Where:
P.E. is the potential energy
k is the spring constant
x is the final length of the spring
L is the natural length of the spring
To find the spring constant, we can use the given information that a 27-N force is required to keep the spring stretched to a length of 30 cm. The spring constant can be calculated using Hooke's Law:
F = k * x
Where:
F is the force applied
k is the spring constant
x is the displacement from the natural length
Let's first calculate the spring constant:
k = F / x = 27 N / 0.06 m = 450 N/m
Now we can calculate the work required to stretch the spring from 24 cm to 27 cm. Let's denote the initial length (L1) as 24 cm and the final length (L2) as 27 cm.
Work (W) = P.E.(L2) - P.E.(L1)
W = 0.5 * k * (L2 - L)^2 - 0.5 * k * (L1 - L)^2
W = 0.5 * (450 N/m) * (0.27 m - 0.24 m)^2 - 0.5 * (450 N/m) * (0.27 m - 0.24 m)^2
W = 0.5 * (450 N/m) * (0.03 m)^2
W = 0.5 * 450 N/m * 0.0009 m^2
W = 0.225 J
Therefore, the work required to stretch the spring from 24 cm to 27 cm is 0.225 Joules.
To find the work required to stretch the spring from 24 cm to 27 cm, we can use the work formula:
W = (1/2)k(x2^2 - x1^2)
where W is the work done, k is the spring constant, x2 is the final displacement, and x1 is the initial displacement.
First, let's calculate the spring constant, k. The formula to find the spring constant is:
k = F / x
where F is the force applied and x is the displacement.
Given that a 27 N force is required to keep the spring stretched to a length of 30 cm (0.30 m), we have:
k = 27 N / 0.30 m
k = 90 N/m
Now, using the work formula with the given values for the initial and final displacements:
W = (1/2)(90 N/m)(0.27 m^2 - 0.24 m^2)
W = (1/2)(90 N/m)(0.729 m^2 - 0.576 m^2)
W = (1/2)(90 N/m)(0.153 m^2)
W = (1/2)(90 N/m)(0.153 m^2)
W = 6.5715 J
Rounding to two decimal places, the work required to stretch the spring from 24 cm to 27 cm is approximately 6.57 J.