The drawing shows a 29.7-kg crate that is initially at rest. Note that the view is one looking down on the top of the crate. Two forces, F1=88N and F2=54N, are applied to the crate, and it begins to move. The angle is 55 degrees. The coefficient of kinetic friction between the crate and the floor is k = 0.324. Determine the (a) magnitude and (b) direction (relative to the x axis) of the acceleration of the crate.
To determine the magnitude and direction of the acceleration of the crate, we need to analyze the forces acting on it.
First, we can determine the net force acting on the crate. The net force can be calculated by finding the vector sum of all the forces acting on the crate.
Net force (F_net) = F1 + F2
F1 = 88N (force applied at an angle of 55 degrees)
F2 = 54N (force applied at an angle of 0 degrees with the x-axis)
To calculate the components of the forces, we can use trigonometry:
F1_x = F1 * cos(55 degrees)
F1_y = F1 * sin(55 degrees)
F2_x = F2 * cos(0 degrees) = F2
F2_y = F2 * sin(0 degrees) = 0
F_net_x = F1_x + F2_x
F_net_y = F1_y + F2_y
Next, we need to consider the force of kinetic friction acting on the crate. The force of kinetic friction can be calculated by multiplying the coefficient of kinetic friction (k) by the normal force (N), which is equal to the weight of the crate.
N = m * g
m = 29.7 kg (mass of the crate)
g = 9.8 m/s^2 (acceleration due to gravity)
Now, we can calculate the magnitude of the force of kinetic friction (F_friction) using the formula:
F_friction = k * N
Finally, we can determine the net force in the x and y directions (F_net_x and F_net_y), taking into account the force of kinetic friction. The net force in the y-direction would be balanced, as there is no vertical acceleration.
F_net_x = F_net_x - F_friction
F_net_y = 0 (balanced forces in the y-direction)
The acceleration of the crate can be calculated using Newton's second law of motion:
a = F_net / m
Now we can substitute the values into the equation and calculate:
a = (F_net_x) / m
This will give us the magnitude of the acceleration.
To determine the direction of the acceleration, we need to find the angle it makes with the x-axis. This can be calculated using the inverse tangent function:
angle = tan^(-1)(F_net_y / F_net_x)
Putting everything together, you can calculate the magnitude and direction of the acceleration of the crate.