A crate starts from rest and slides 8.35 m down a ramp. When it reaches the bottom it is traveling at a speed of 5.25 m/s. The ramp makes an angle of 20.0o with the horizontal.
b) What is the coefficient of kinetic friction between the crate and the ramp?
To find the coefficient of kinetic friction between the crate and the ramp, we can use the following steps:
1. Start by determining the acceleration of the crate as it slides down the ramp using the equation:
v^2 = u^2 + 2as
Here, v = final velocity = 5.25 m/s (given)
u = initial velocity = 0 m/s (as it starts from rest)
a = acceleration
s = distance traveled = 8.35 m (given)
Rearranging the equation, we have:
a = (v^2 - u^2) / (2s)
= (5.25^2 - 0^2) / (2*8.35)
2. Next, find the component of the crate's weight that is parallel to the ramp. This is done by multiplying the weight of the crate by the cosine of the angle of the ramp:
Weight_parallel = Weight * cos(angle)
= m * g * cos(angle)
Here, m = mass of the crate
g = acceleration due to gravity (approximately 9.8 m/s^2)
angle = 20.0 degrees (given)
3. Use the formula for net force to find the net force acting on the crate in the direction of motion:
Net force = m * a
= Weight_parallel - frictional force
The frictional force acting on the crate is given by:
Frictional force = coefficient of friction * Normal force
The normal force is equal to the weight of the crate perpendicular to the ramp:
Normal force = Weight * sin(angle)
= m * g * sin(angle)
4. Substituting the values into the net force equation:
m * a = m * g * cos(angle) - coefficient of friction * m * g * sin(angle)
The mass of the crate (m) cancels out from both sides:
a = g * cos(angle) - coefficient of friction * g * sin(angle)
5. Now we can solve for the coefficient of kinetic friction:
coefficient of friction = (g * cos(angle) - a) / (g * sin(angle))
Substituting the known values:
coefficient of friction = (9.8 * cos(20.0) - a) / (9.8 * sin(20.0))
6. Finally, plug in the value calculated for acceleration (a) in step 1 to find the coefficient of kinetic friction.
coefficient of kinetic friction = (9.8 * cos(20.0) - a) / (9.8 * sin(20.0))
PE loss = KE gain + work againat friction
M g *8.35*sin20
= (M/2)*Vfinal^2 + 8.35*M*g*cos20*mu_k
Cancel the M's and solve for mu_k