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 = 29 kg and the coefficient of static friction between the crate and the floor is 0.64.

help me solve this. thanks

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

Let's break down the forces acting on the crate:

1. Weight (W = m * g): The weight of the crate acts vertically downward, and its magnitude is given by the mass (m) of the crate multiplied by the acceleration due to gravity (g ≈ 9.8 m/s²).

2. Normal Force (N): The normal force is the force exerted by the floor on the crate, perpendicular to the surface. Since the crate is on a rough floor, the normal force is equal to the weight of the crate (N = W = m * g).

3. Applied Force (F): The force you apply to push the crate at an angle of 21° below the horizontal.

4. Frictional Force (f): The frictional force opposes the motion of the crate and can be calculated as the product of the coefficient of static friction (μs) and the normal force (N).

Since the crate is on the verge of moving, the applied force (F) must overcome the static frictional force (f). The maximum static frictional force can be found using the equation:

f = μs * N

Now we can calculate the force necessary to start the crate moving:

f = μs * N
f = 0.64 * (m * g)

Next, we need to resolve the applied force (F) into its horizontal and vertical components:

F(horizontal) = F * cos(θ)
F(vertical) = F * sin(θ)

where θ is the angle of 21° below the horizontal.

Now, the net force along the horizontal direction is given by:

Net Force = F(horizontal) - f

To start the crate moving, the net force must be greater than zero. Therefore:

Net Force > 0
F(horizontal) - f > 0
F * cos(θ) - 0.64 * (m * g) > 0

Now, we can substitute the known values into the equation and solve for F:

F * cos(21°) - 0.64 * (29 kg * 9.8 m/s²) > 0

Simplifying the equation:

F * 0.9272 - 179.8784 > 0
F * 0.9272 > 179.8784
F > 179.8784 / 0.9272

F > 194.091 N

Therefore, the force necessary to start the crate moving is approximately 194.091 N.

To solve this problem, we need to consider the forces acting on the crate and find the force necessary to overcome the static friction and start the crate moving.

First, let's draw a free-body diagram of the forces on the crate:

1. Gravity (mg): This force acts vertically downward and its magnitude is given by the mass of the crate (m = 29 kg) multiplied by the acceleration due to gravity (g ≈ 9.8 m/s²).

2. Normal force (N): This force acts perpendicular to the surface of the rough floor and counteracts the force of gravity. It has the same magnitude as the weight of the crate (mg), but opposite in direction.

3. Applied force (F): This is the force you are applying to push the crate. It acts at an angle of 21° below the horizontal.

4. Frictional force (f): This force acts parallel to the surface of the rough floor and opposes the motion of the crate. The static frictional force must be overcome to start the crate moving.

Now, let's write down the equations for the forces in the x and y directions:

In the y-direction:
N - mg = 0
N = mg

In the x-direction:
Fcos(21°) - f = 0

The static frictional force (f) can be expressed as the coefficient of static friction (μs) multiplied by the normal force (N):
f = μsN

Substituting the value of N from the equation in the y-direction, we get:
f = μsmg

Now, we know that the static frictional force must be overcome to start the crate moving, so it can be equal to the maximum static frictional force (f_max). The formula for the maximum static frictional force is:
f_max = μsN

Substituting the value of N, we get:
f_max = μsmg

Now, we can equate the equations for the x-direction and the maximum static frictional force:
Fcos(21°) - μsmg = 0

Solving for the applied force (F), we get:
F = μsmg / cos(21°)

Plugging in the values given:
F = (0.64)(29 kg)(9.8 m/s²) / cos(21°)

Calculating this expression will give you the force necessary to start the crate moving.