A 287-kg crate is being pushed across a horizontal floor by a force P that makes an angle of 25.6 ° below the horizontal. The coefficient of kinetic friction is 0.236. What should be the magnitude of P, so that the net work done by it and the kinetic frictional force is zero?
To find the magnitude of force P, such that the net work done by it and the kinetic frictional force is zero, we need to analyze the forces acting on the crate.
1. Determine the weight of the crate:
The weight (W) of the crate is given by W = m * g, where m is the mass of the crate and g is the acceleration due to gravity. In this case, m = 287 kg and g = 9.8 m/s^2. So, W = 287 kg * 9.8 m/s^2 = 2813.6 N.
2. Determine the normal force:
The normal force (N) is the force exerted by the floor on the crate, perpendicular to the horizontal surface. Since the crate is on a horizontal surface, the normal force is equal in magnitude and opposite in direction to the weight of the crate. So, N = 2813.6 N.
3. Determine the frictional force:
The frictional force (f) can be calculated using the equation f = μ * N, where μ is the coefficient of kinetic friction. In this case, the coefficient of friction (μ) is given as 0.236 and the normal force (N) is 2813.6 N. So, f = 0.236 * 2813.6 N = 664.6336 N.
4. Determine the horizontal component of the force P:
The force P has two components - one in the horizontal direction and one in the vertical direction. We need to find the horizontal component of P.
The horizontal component of P (P_x) can be determined using the equation P_x = P * cos(θ), where θ is the angle that P makes with the horizontal. In this case, θ = 25.6°. So, P_x = P * cos(25.6°).
5. Equating work done by P and frictional force:
To have zero net work done, the work done by the horizontal component of P should be equal and opposite to the work done by the frictional force, i.e., P_x * d = -f * d, where d is the displacement.
Since the work done is equal to force multiplied by displacement, we can rewrite the equation as:
P_x = -f
Plugging in the values we calculated earlier, we have:
P * cos(25.6°) = -664.6336 N
Now, we can solve for the magnitude of P:
P = -664.6336 N / cos(25.6°)
Calculating the value, we get:
P ≈ -743.281 N
Since force is a vector quantity, the negative sign indicates the direction opposite to the direction of P. So, the magnitude of P should be approximately 743.281 Newtons.