A box of unknown mass is placed at rest at the top of a 3m long ramp that is inclined at 30 degrees, and it slides to the bottom in 2s. What is the acceleration of the box, what is the coeff of kinetic friction between the box and the plank?
To find the acceleration of the box, we can use the equations of motion. The first step is to determine the final velocity of the box when it reaches the bottom of the ramp.
Using the equation of motion: v = u + at, where:
v = final velocity (which is 0 since the box comes to rest at the bottom)
u = initial velocity (which is unknown)
a = acceleration (which we want to find)
t = time (which is given as 2 seconds)
Since the box starts from rest, the initial velocity (u) is 0. So, the equation becomes:
0 = 0 + a * 2
0 = 2a
Therefore, the acceleration of the box is 0 m/s^2.
Now, to find the coefficient of kinetic friction between the box and the plank, we need to consider the forces acting on the box along the incline.
There are two main forces at play:
1. The component of the box's weight parallel to the incline, which is ma,
2. The force of kinetic friction, which is opposite in direction to the box's motion.
Using Newton's second law, F = ma, we can write the equation for the forces along the incline:
ma = m * g * sin(θ) - μ * m * g * cos(θ)
Where:
m = mass of the box (unknown)
g = acceleration due to gravity (9.8 m/s^2)
θ = angle of the incline (30 degrees)
μ = coefficient of kinetic friction (unknown)
Since the mass (m) cancels out on both sides, we can simplify the equation as follows:
a = g * sin(θ) - μ * g * cos(θ)
Substituting the given values:
a = 9.8 * sin(30) - μ * 9.8 * cos(30)
We can solve for μ by rearranging the equation:
μ = (9.8 * sin(30) - a) / (9.8 * cos(30))
Now, substitute the acceleration (a = 0) to find the coefficient of kinetic friction (μ):
μ = (9.8 * sin(30) - 0) / (9.8 * cos(30))
μ = sin(30) / cos(30)
μ = √3 / 2
Therefore, the coefficient of kinetic friction between the box and the plank is (√3 / 2).