A man pushing a crate of mass m = 92.0 kg at a speed of v = 0.860 m/s encounters a rough horizontal surface of length = 0.65 m as in the figure below. If the coefficient of kinetic friction between the crate and rough surface is 0.355 and he exerts a constant horizontal force of 300 N on the crate, find the following.
Find the magnitude and direction of the net force on the crate while it is on the rough surface.
Well, well, well, looks like someone is trying to push their luck (and a crate)! Let's see what we can do here.
To find the magnitude of the net force on the crate, we first need to calculate the frictional force acting on it. The formula for frictional force is given by:
frictional force = coefficient of kinetic friction * normal force
Now, the normal force is perpendicular to the surface and equal in magnitude and opposite in direction to the weight of the crate (mg). So, the normal force can be calculated as follows:
normal force = mass * gravity
Using the given mass, m = 92.0 kg, and the acceleration due to gravity, g = 9.8 m/s^2, we can find the normal force.
normal force = 92.0 kg * 9.8 m/s^2
Once we have the normal force, we can determine the frictional force:
frictional force = 0.355 * normal force
Now, since the man is exerting a constant horizontal force of 300 N on the crate, the net force can be calculated by subtracting the frictional force from the applied force:
net force = applied force - frictional force
Finally, the direction of the net force is in the direction opposite to the applied force. We're assuming the applied force is to the right, so the net force will be to the left.
So, let's crunch the numbers and see what we get! (drumroll, please)
To find the magnitude and direction of the net force on the crate while it is on the rough surface, we need to consider the forces acting on the crate.
1. Gravitational force (Weight): The weight of the crate is given by the formula:
Weight = mass * acceleration due to gravity
Given that the mass of the crate is m = 92.0 kg, and the acceleration due to gravity is approximately 9.8 m/s^2, the weight of the crate can be calculated as follows:
Weight = 92.0 kg * 9.8 m/s^2
2. Normal force (N): The normal force acts perpendicular to the surface and balances the weight of the crate. Therefore, it can be calculated as:
Normal force = Weight
3. Frictional force: The frictional force can be calculated using the formula:
Frictional force = coefficient of kinetic friction * Normal force
Given that the coefficient of kinetic friction is μ = 0.355, and we have calculated the Normal force in the previous step, the frictional force can be calculated as follows:
Frictional force = 0.355 * Normal force
4. Applied force: The person is exerting a constant horizontal force on the crate. Given that the force applied by the person is F = 300 N, this force acts in the direction of motion of the crate.
The net force on the crate is the vector sum of all the forces acting on it. Since we're dealing with forces, we must consider both magnitude and direction. The direction of the net force will determine whether the crate accelerates or decelerates.
To find the magnitude and direction of the net force, add up the magnitudes of the forces acting in the horizontal direction and subtract any forces acting in the opposite direction.
Net force = Applied force - Frictional force
To determine the direction of the net force, consider the sign of the result. If it is positive, the force is in the positive direction (in the direction of motion), and if it is negative, the force is in the negative direction (opposite to the direction of motion).
To find the magnitude and direction of the net force on the crate while it is on the rough surface, we need to consider the forces acting on the crate.
First, let's determine the force of friction acting on the crate. The force of friction can be calculated using the equation:
Frictional Force = coefficient of friction * normal force
where the normal force is the force exerted by the surface perpendicular to the crate. In this case, the normal force is equal to the weight of the crate, which can be calculated as:
Normal Force = mass * gravity
where gravity is the acceleration due to gravity, approximately 9.8 m/s^2.
Given that the mass of the crate is 92.0 kg, the coefficient of kinetic friction is 0.355, and the acceleration due to gravity is 9.8 m/s^2, we can calculate the normal force and the force of friction.
Normal Force = mass * gravity = 92.0 kg * 9.8 m/s^2 = 901.6 N
Frictional Force = coefficient of friction * normal force = 0.355 * 901.6 N = 320.04 N
Now, let's consider the horizontal force applied by the man, which is 300 N. Since the crate is being pushed horizontally, this force is equal in magnitude but opposite in direction to the net force on the crate.
The net force acting on the crate can be calculated using the equation:
Net Force = Applied Force - Frictional Force
Net Force = 300 N - 320.04 N = -20.04 N
The negative sign indicates that the net force is acting in the opposite direction to the applied force.
Therefore, the magnitude of the net force on the crate while it is on the rough surface is approximately 20.04 N, and the direction is opposite to the applied force (in the direction opposite to the motion of the crate).
The net force in the "rough patch" is the pushing force minus the kinetic friction force.
Fnet = 300 - M*g*0.355 = -20 N
The crate will decelerate at a rate
a = Fnet/M = 3.38 m/s^2
It will continue moving for a time
t = Vo/a = 0.254 seconds, at which time it will stop after traveling 0.11 meters
The applied force will not be enough to move it farther.