For a particular nonlinear spring the relationship between the magnitude of the applied for F and the resultant displacement x from equilibrium is given by the equation F=kx^2. What is the amount of work done by stretching the spring a distance x_0?
a)kx_0^2
b) 1/2 kx_0
c)1/2 kx_0^3
d) 1/3 kx_0^2
e)1/3 kx_0^3
W=1/2kx_i^2 -1/2kx_f^2 but I don't know what I would do with this equation. Can you please explain how I would solve this problem?
To get the work, you have to integrate the force as a function of x from zero displacement to Xo
(Integral of) k x^2 dx from 0 to Xo = (1/3) k (Xo)^3
Therefore, the answer is e) 1/3 kx_0^3
Well, solving this problem involves a bit of math, but don't worry, I'm here to make it fun!
First, let's understand what we need to do. We want to find the amount of work done by stretching the spring a distance x₀, given the relationship F = kx².
Now, to calculate work, we know that W = ∫F dx, where F is the force and dx is the displacement. In our case, F = kx², and we need to integrate this with respect to x from 0 to x₀.
So, let's get mathematical for a moment. The integral becomes:
W = ∫kx² dx from 0 to x₀
To solve this integral, we can use the power rule. Integrating kx² gives us (1/3)kx³.
Now, we can evaluate the integral from 0 to x₀:
W = [(1/3)kx³] evaluated at x₀ and 0
Plugging in x₀ for x, we get:
W = (1/3)k(x₀)³ - (1/3)k(0)³
As you can see, the second term becomes 0, and we're left with:
W = (1/3)k(x₀)³
So, the amount of work done by stretching the spring a distance x₀ is (1/3)k(x₀)³.
Now, that wasn't too scary, was it? Just a little bit of integration, and you've got your answer. And remember, math is always more fun with a smile!
To solve this problem, you can use the formula for work done by a variable force. The formula is:
W = ∫F dx
In this case, the force F is given by F = kx^2, where k is the spring constant and x is the displacement from equilibrium. We want to find the amount of work done by stretching the spring a distance x_0, so we need to integrate the force from 0 to x_0.
Substituting F = kx^2, the equation becomes:
W = ∫(kx^2) dx
Now we can integrate this equation. The integral of x^2 with respect to x is (1/3) x^3. So, we have:
W = ∫kx^2 dx = k ∫x^2 dx = k (1/3) x^3
Next, we will evaluate this integral from 0 to x_0:
W = k (1/3) [x^3] from 0 to x_0
To evaluate the integral at the limits, substitute x_0 and 0 into the expression:
W = k (1/3) [x_0^3] - k (1/3) [0^3]
Since 0^3 is zero, we can simplify the expression:
W = (1/3) k (x_0^3)
Therefore, the amount of work done by stretching the spring a distance x_0 is (1/3) k (x_0^3).
So, the correct answer is e) 1/3 k x_0^3.
To solve this problem, you need to calculate the work done by stretching the spring a distance x₀. The equation you mentioned, W = (1/2)kxᵢ² - (1/2)kx_f², is the formula for calculating work using the integration method.
The work done by stretching the spring can be found by integrating the force equation, F = kx², from zero displacement (x = 0) to the given displacement x₀. The integral is the area under the force-displacement curve, which represents the work done.
To integrate the equation F = kx² with respect to displacement x, you can use the power rule of integration. The power rule states that an integral of the form ∫axⁿ dx, where a is a constant and n is any real number except -1, is given by (a/(n+1))x^(n+1) + C.
Applying the power rule to the force equation, you get:
∫kx² dx = (k/3)x³ + C
To find the definite integral from x = 0 to x = x₀, you subtract the result at x = 0 from the result at x = x₀:
W = [(k/3)x₀³ + C] - [(k/3)(0)³ + C]
W = (k/3)x₀³ - (k/3)(0)
W = (k/3)x₀³
So, the amount of work done by stretching the spring a distance x₀ is (k/3)x₀³.
Looking at the answer choices, e) 1/3 kx₀³ matches the result we obtained, indicating that e) is the correct answer.