To move a large crate across a rough floor, you push on it with a force F at an angle of 21° below the horizontal, as shown in the figure. Find the force necessary to start the crate moving, given that the mass of the crate is m = 36 kg and the coefficient of static friction between the crate and the floor is 0.67.

I am not certain which way the force is directed: I assume downward.

force normal=mg+F*sinTheta
frictionforce=mu*forcenormal

to start the motion, F*cosTheta = friction force, solve for F. It's just algebra.

236.4+.24F

To find the force necessary to start the crate moving, we need to consider the forces acting on the crate and the equilibrium condition.

First, let's identify the forces acting on the crate:
1. Weight (W) acting vertically downward with a magnitude of W = m * g, where m is the mass of the crate and g is the acceleration due to gravity.
2. Normal force (N) exerted by the floor on the crate, perpendicular to the plane of motion.
3. Force of static friction (f_s) opposing the motion of the crate.

Since the crate is at rest, the net force in the horizontal direction should be zero, according to the equilibrium condition.

Now, let's analyze the forces in the horizontal direction:
1. The horizontal component of the applied force (F_h) is F * cos(21°).
2. The force of static friction can oppose the applied force to prevent motion. The maximum force of static friction is given by f_s_max = coefficient of static friction * N.

Since the crate is just on the verge of moving, the force of static friction should be equal to the maximum static friction force, according to the question.

Setting up the equation based on the equilibrium condition:
F_h - f_s_max = 0

Substituting the values:
F * cos(21°) - coefficient of static friction * N = 0

To find the normal force, we need to consider the vertical forces.
The vertical component of the applied force (F_v) is F * sin(21°).
So, N - W - F_v = 0

Substituting the values:
N - m * g - F * sin(21°) = 0

We can solve these two equations simultaneously to find the force necessary to start the crate moving.

1. Calculate the weight:
W = m * g = 36 kg * 9.8 m/s² = 352.8 N

2. Calculate the maximum force of static friction:
f_s_max = coefficient of static friction * N = 0.67 * N

3. Substitute the values into the equilibrium equation for the vertical forces:
N - W - F * sin(21°) = 0

N - 352.8 N - F * sin(21°) = 0

4. Solve for N:
N = 352.8 N + F * sin(21°)

5. Substitute the value of N into the equilibrium equation for the horizontal forces:
F * cos(21°) - coefficient of static friction * N = 0

F * cos(21°) - 0.67 * (352.8 N + F * sin(21°)) = 0

6. Solve for F:
F * cos(21°) - 0.67 * 352.8 N - 0.67 * F * sin(21°) = 0

F * (cos(21°) - 0.67 * sin(21°)) = 0.67 * 352.8 N

F = (0.67 * 352.8 N) / (cos(21°) - 0.67 * sin(21°))

Using a calculator, we can evaluate the right-hand side of the equation to find the force necessary to start the crate moving.