The drawing shows a 25.0-kg crate that is initially at rest. Note that the view is one looking down on the top of the crate. Two forces, F1 and F2, are applied to the crate, and it begins to move. The coefficient of kinetic friction between the crate and the floor is µk = 0.350. Determine the magnitude and direction (relative to the x axis) of the acceleration of the crate.
We can use Newton's second law, which states that the net force on an object is equal to its mass times its acceleration:
ΣF = ma
We need to find the net force on the crate. There are two forces acting on it, F1 and F2:
ΣF = F1 + F2
The direction of the acceleration will be in the direction of the net force.
To find F1 and F2, we can use trigonometry. The angle between F1 and the x-axis is 30 degrees, and the angle between F2 and the x-axis is 60 degrees. We know the magnitude of F1 is 45.0 N, and the magnitude of F2 is 60.0 N.
F1x = F1 cos 30 = 45.0 N cos 30 = 38.9 N
F1y = F1 sin 30 = 45.0 N sin 30 = 22.5 N
F2x = F2 cos 60 = 60.0 N cos 60 = 30.0 N
F2y = F2 sin 60 = 60.0 N sin 60 = 51.96 N
The forces in the y-direction will cancel each other out, since they are in opposite directions. So we only need to consider the forces in the x-direction:
ΣFx = F1x + F2x = 38.9 N + 30.0 N = 68.9 N
Now we can find the net force:
ΣF = μkN + ΣFx
where N is the normal force, which is equal in magnitude to the weight of the crate:
N = mg = (25.0 kg)(9.81 m/s^2) = 245 N
ΣF = (0.350)(245 N) + 68.9 N = 151.5 N
The direction of the acceleration is in the direction of the net force, which is 30 degrees above the x-axis (since F1 is at a 30 degree angle). Therefore:
a = ΣF/m = (151.5 N)/(25.0 kg) = 6.06 m/s^2
θ = 30 degrees above the x-axis
So the acceleration of the crate is 6.06 m/s^2, at a 30 degree angle above the x-axis.
To determine the magnitude and direction of the acceleration of the crate, we need to consider the forces acting on it.
1. The force F1 is applied horizontally to the right.
2. The force F2 is applied at an angle relative to the x-axis.
Since the crate is initially at rest, the only force that can cause it to move is the horizontal component of F2. This component, called F2x, can be calculated using trigonometry:
F2x = F2 * cos θ
Now let's identify the forces opposing the motion of the crate. The only force in this case is the force of kinetic friction (fk). The magnitude of fk can be calculated using the formula:
fk = μk * N
where μk is the coefficient of kinetic friction and N is the normal force.
Since the crate is on a horizontal surface, the normal force is equal to the weight of the crate:
N = m * g
where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s²).
Now, let's substitute these values into the equations:
F2x = F2 * cos θ
fk = μk * N = μk * (m * g)
Now we can calculate the acceleration (a) using Newton's second law:
ΣF = ma
Since F1 and F2x are both horizontal and in the same direction, we have:
ΣF = F1 + F2x - fk
Substituting the calculated values:
ma = F1 + (F2 * cos θ) - (μk * m * g)
Finally, we can solve for the magnitude and direction of the acceleration (a) by rearranging the equation:
a = (F1 + F2x - μk * m * g) / m
Now we have the equation to calculate the magnitude and direction of the acceleration of the crate. Plug in the given values for F1, F2, θ, μk, m, and g to find the answer.