A 12-kg crate rests on a horizontal surface and a boy pulls on it with a force that is 30 degree below the horizontal. If the coefficient of static friction is 0.40, the minimum magnitude force he needs to start the crate moving is:

about 71N

Well, if the crate is resting, it seems like it's enjoying a good coffee break on that horizontal surface. But wait, here comes the boy, ready to disrupt the crate's peaceful slumber!

Now, the boy pulls on the crate with a force that is 30 degrees below the horizontal. I hope he doesn't accidentally fall down while doing that! Anyway, let's get back to the question.

In order to start the crate moving, the force exerted by the boy must overcome the static friction acting on the crate. And the magnitude of this force can be calculated using the formula:

Fs = μs * N

Where Fs is the force of static friction, μs is the coefficient of static friction, and N is the normal force.

Since we're dealing with a horizontal surface, the normal force is equal to the weight of the crate, which is given by:

N = m * g

Where m is the mass of the crate and g is the acceleration due to gravity.

So, substituting the values into the formulas, we have:

Fs = μs * N

Fs = 0.40 * (12 kg * 9.8 m/s^2)

Now, go ahead and crunch the numbers to find out the minimum magnitude force the boy needs to start the crate moving.

To determine the minimum force needed to start the crate moving, we need to consider the forces acting on the crate.

The force being applied by the boy is at an angle of 30 degrees below the horizontal. We can split this force into its horizontal and vertical components.

The horizontal component of the force can be calculated using the formula:

F_horizontal = F * cos(θ)

where F is the magnitude of the force and θ is the angle.

Given that the force is being applied at an angle of 30 degrees below the horizontal, we have:

F_horizontal = F * cos(30) = F * √3/2

Now, we need to consider the static friction acting on the crate. The maximum static friction force can be calculated using the formula:

F_friction = μ_s * N

where μ_s is the coefficient of static friction and N is the normal force.

The normal force is equal to the weight of the crate, which can be calculated using:

N = m * g

where m is the mass of the crate and g is the acceleration due to gravity.

Given that the mass of the crate is 12 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we have:

N = 12 kg * 9.8 m/s^2 = 117.6 N

Substituting the values for the coefficient of static friction and the normal force, we have:

F_friction = 0.40 * 117.6 N = 47.04 N

In order to start the crate moving, the applied force must be greater than or equal to the static friction force. Therefore, the minimum magnitude force needed to start the crate moving is:

Minimum force = F_friction = 47.04 N

So, the minimum magnitude force the boy needs to start the crate moving is 47.04 N.

To find the minimum magnitude force needed to start the crate moving, we need to consider the forces acting on the crate. The force applied by the boy can be resolved into two components: one perpendicular to the surface and one parallel to the surface.

First, let's find the perpendicular component of the force applied by the boy. We can use trigonometry to determine this component:

Perpendicular component = Force applied * cos(θ)

where θ is the angle below the horizontal (30 degrees).

Perpendicular component = Force applied * cos(30 degrees)

Next, let's find the maximum static friction force that opposes the motion of the crate. The maximum static friction force can be calculated using the equation:

Maximum static friction force = coefficient of static friction * normal force

The normal force is the force exerted by the surface on the crate, which is equal to the weight of the crate:

Normal force = mass * acceleration due to gravity

Normal force = 12 kg * 9.8 m/s^2

Now, we can calculate the maximum static friction force:

Maximum static friction force = 0.40 * (12 kg * 9.8 m/s^2)

Finally, the minimum magnitude force needed to start the crate moving is equal to the maximum static friction force.

So, the answer is the maximum static friction force calculated above.

m*g = 12kg * 9.8N/kg = 117.6 N. = Wt. of

the crate.

Fs = u*mg = 0.4 * 117.6 = 47.04 N. =
Force of static friction.

F*cos(-30) - Fs = m*a
0.8669F - 47.04 = m*0 = 0
0.866F = 47.04
F = 54.32 N.