When a particle is located a distance x meters from the origin, a force of cos(x/4) newtons acts on it. How much work W is done in moving the particle from x = 1 to x = 3?
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.)
To calculate the work done W in moving the particle from x = 1 to x = 3, we need to integrate the force over this range.
The work done by a force F(x) when moving a particle over a distance dx is given by the equation:
dW = F(x) dx
Since we have a function F(x) = cos(x/4), we can rewrite the equation as:
dW = cos(x/4) dx
To find the total work done, we need to integrate this equation from x = 1 to x = 3:
W = ∫[1 to 3] cos(x/4) dx
To evaluate this integral, we can use the substitution u = x/4:
du = (1/4) dx
dx = 4 du
Now we can rewrite the integral in terms of u:
W = ∫[1 to 3] cos(u) (4 du)
W = 4 ∫[1 to 3] cos(u) du
Using the integral of the cosine function:
W = 4 [sin(u)] [1 to 3]
W = 4 (sin(3) - sin(1))
To find the work done in moving the particle from x = 1 to x = 3, we need to integrate the force function over the distance traveled.
The work done, W, is given by the formula:
W = ∫[x1, x2] F(x) dx
Where F(x) is the force function and [x1, x2] is the interval over which the particle is moved.
In this case, F(x) = cos(x/4) newtons and the interval is from x = 1 to x = 3.
So, we have:
W = ∫[1, 3] cos(x/4) dx
To solve this integral, we can use the substitution method.
Let u = x/4, then du = dx/4.
When x = 1, u = 1/4, and when x = 3, u = 3/4.
Now, we can rewrite the integral as:
W = ∫[1/4, 3/4] 4 cos(u) du
W = 4 ∫[1/4, 3/4] cos(u) du
W = 4 [sin(u)] [1/4, 3/4] (using the integral of cos(u) = sin(u))
W = 4 [sin(3/4) - sin(1/4)]
Now we can calculate the value of W by plugging in the values:
W ≈ 4 [0.6816 - 0.2474]
W ≈ 1.3744 Newton-meters
So, the work done in moving the particle from x = 1 to x = 3 is approximately 1.3744 Newton-meters.