A 250-kg crate is being pushed across a horizontal floor by a force P that makes an angle of 30.8 ° below the horizontal. The coefficient of kinetic friction is 0.224. What should be the magnitude of P, so that the net work done by it and the kinetic frictional force is zero?
Thus far, I know Pcos(30.8) = fk, where fk= (0.224)(mg+Psin(30.8)). I am not sure, however, how I can solve for the magnitude of the P.
no, if the force is pushing up, the term in vertical forces is mg-PSin(angle)
P cosTheta=.224(mg-PsinTheta)
Now, you know that for net work to be zero, friction has to be zero. which means mg-PsinTheta=0
solve for P
To solve for the magnitude of P, we need to set up an equation where the net work done by P and the kinetic frictional force is zero.
First, let's break down the forces acting on the crate. We have:
1. The force P, which is being applied at an angle of 30.8° below the horizontal (Pcos(30.8) is the horizontal component and Psin(30.8) is the vertical component).
2. The force of kinetic friction, fk, which opposes the motion of the crate.
The work done by P is given by W_P = force P × displacement, and the work done by the kinetic frictional force is given by W_fk = force fk × displacement.
Since the net work is zero, we can set up the equation:
W_P + W_fk = 0
Substituting the formulas for work, we have:
(Pcos(30.8) × displacement) + (fk × displacement) = 0
Now, we can incorporate the equation you provided, fk = (0.224)(mg + Psin(30.8)):
Pcos(30.8) × displacement + (0.224)(mg + Psin(30.8)) × displacement = 0
To solve for P, we need to isolate it in the equation. Here's how to rearrange the equation:
Pcos(30.8) × displacement + 0.224mg × displacement + 0.224Psin(30.8) × displacement = 0
P(cos(30.8) + 0.224sin(30.8)) × displacement + 0.224mg × displacement = 0
P = (-0.224mg × displacement) / (cos(30.8) + 0.224sin(30.8))
To find the displacement, you would need additional information, such as the distance the crate is being pushed. Substitute the displacement value into the equation above along with the given values for m, g, and the coefficient of kinetic friction, to calculate the magnitude of P.