The magnitude of a horizontal force acting on a point mass M on a frictionless horizontal surface is F=bx^2 where b is a constant and x is the distance in meters (box starts from rest). Find the speed of the point mass after x=L. Solve for b=0.1, M=1 Kg, L=1 m. Your answer in X.XX m/s
Question
The magnitude of a horizontal force acting on a point mass M on a frictionless horizontal surface is F=bx^2 where b is a constant and x is the distance in meters (box starts from rest). Find the speed of the point mass after x=L. Solve for b=0.1, M=1 Kg, L=1 m. Your answer in X.XX m/s
Solution
The problem involves a force that varies with the square of the distance, and we need to find the speed of the point mass after it has moved a certain distance.
We can start by using the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy.
The work done by a force F over a distance L is given by the integral of F dx from 0 to L. In this case, F = bx^2, so we need to compute the integral of bx^2 dx from 0 to L.
∫ from 0 to L of bx^2 dx = b/3 * x^3 evaluated from 0 to L = b/3 * L^3.
The work done is then equal to the change in kinetic energy, which is 1/2 * M * v^2, where v is the final speed.
Setting these equal gives us:
b/3 * L^3 = 1/2 * M * v^2.
We can solve this equation for v:
v = sqrt((2 * b * L^3) / (3 * M)).
Substituting the given values b = 0.1, M = 1 kg, and L = 1 m, we get:
v = sqrt((2 * 0.1 * 1^3) / (3 * 1)) = sqrt(0.2/3) = sqrt(0.0667) = 0.26 m/s.
So, the speed of the point mass after it has moved a distance of 1 m is approximately 0.26 m/s.
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