A spring has a natural length of 24 cm. If a 20-N force is required to keep it stretched to a length of 40 cm, how much work W is required to stretch it from 24 cm to 32 cm? (Round your answer to two decimal places.
Well, W-work hard and I'll provide you with the answer!
To find the work required to stretch the spring from 24 cm to 32 cm, we need to calculate the difference in potential energy stored in the spring.
First, let's calculate the initial potential energy when the spring is at its natural length of 24 cm. The formula for potential energy stored in a spring is given by:
PE = (1/2)kx^2
where PE is the potential energy, k is the spring constant, and x is the displacement from the natural length.
Since the natural length of the spring is 24 cm, the initial displacement is 0 cm. Therefore, the initial potential energy is also 0.
Now, let's calculate the final potential energy when the spring is stretched to 32 cm. The displacement is 32 cm - 24 cm = 8 cm.
Given that a force of 20 N is required to keep the spring stretched to 40 cm, we can use Hooke's Law to find the spring constant:
F = kx
20 N = k*(40 cm - 24 cm)
20 N = k*16 cm
k = 20 N / 16 cm
k = 1.25 N/cm
Using this spring constant, we can calculate the final potential energy:
PE = (1/2) * (1.25 N/cm) * (8 cm)^2
PE = 4 Ncm
Therefore, the work required to stretch the spring from 24 cm to 32 cm is equal to the difference in potential energy, which is 4 Ncm.
But hey, don't worry, I won't spring any more math problems on you!
To find the work required to stretch the spring from 24 cm to 32 cm, we can use the equation:
W = 0.5 * k * (x2^2 - x1^2)
where W is the work done, k is the spring constant, and x2 and x1 are the final and initial displacements from the natural length, respectively.
Given that the spring has a natural length of 24 cm, the initial displacement (x1) is 8 cm (32 cm - 24 cm).
To find the spring constant (k), we can use Hooke's Law:
F = k * x
where F is the force applied to the spring and x is the displacement from the natural length.
From the given information, we know that a 20-N force is required to stretch the spring to a length of 40 cm. The displacement (x) in this case is 16 cm (40 cm - 24 cm).
Using Hooke's Law, we can calculate the spring constant (k):
k = F / x = 20 N / 16 cm = 1.25 N/cm
Now we can substitute the values into the work equation:
W = 0.5 * k * (x2^2 - x1^2)
W = 0.5 * 1.25 N/cm * ((32 cm)^2 - (8 cm)^2)
Calculating this expression:
W = 0.5 * 1.25 N/cm * (1024 cm^2 - 64 cm^2)
W = 0.5 * 1.25 N/cm * 960 cm^2
W = 600 N * cm = 6 J
Therefore, the work required to stretch the spring from 24 cm to 32 cm is 6 Joules (J).
To find the work required to stretch the spring from 24 cm to 32 cm, we need to calculate the area under the force-extension curve.
First, let's find the spring constant, k, using Hooke's Law. Hooke's Law states that the force required to stretch or compress a spring is directly proportional to the extension or compression of the spring.
We know that a force of 20 N is required to stretch the spring from its natural length of 24 cm to a length of 40 cm. The extension, Δx, is given by:
Δx = final length - initial length
Δx = 40 cm - 24 cm
Δx = 16 cm
Now we can calculate the spring constant, k, using Hooke's Law:
F = kΔx
Rearranging the equation gives us:
k = F / Δx
k = 20 N / 16 cm
k = 20 N / 0.16 m [Convert cm to m by dividing by 100]
k = 125 N/m
Now we can calculate the work, W, required to stretch the spring from 24 cm to 32 cm. The work is equal to the area under the force-extension curve.
The work done on the spring can be calculated using the formula:
W = (1/2)k(Δx^2)
where k is the spring constant and Δx is the change in length.
Δx = 32 cm - 24 cm
Δx = 8 cm
Converting cm to m:
Δx = 0.08 m
Plugging in the values:
W = (1/2)(125 N/m)(0.08 m)^2
W = (1/2)(125 N/m)(0.0064 m^2)
W = (1/2)(0.8 N)(0.0064 m^2)
W = 0.5 * 0.0064 J
W = 0.0032 J
So, the work required to stretch the spring from 24 cm to 32 cm is approximately 0.0032 Joules.
the spring constant (k) is ... 20 N / (40 - 24) cm = 1.25 N/cm = 125 N/m
work = 1/2 k x^2 = 1/2 * 125 * [(.32 - .24) m]^2 = ? Joules