Whe coefficient of kinetic frictio.
A 300-kg crate is placed on an adjustable inclined plane. As one end of the incline is raised, the crate begins to move downward. If the crate slides down the plane with an acceleration of 0.70 m/s2 , what is the coefficient of kinetic friction between the ramp and crate?
We assume the crate is inclined at an angle ,x to the horizontal. Let us assume the mass of the crate m kg.
Now the force acting on the crate are: (i) weight force of the crate (ii), the force along the plane of the crate due to its acceleration, a. (iii) th friction opposing the force of accelatiing movement.
Resolving the weight force along and vertical to the surface of the plane we get:
Force along the plane = mg sinx, where g is the acceleration due to gravitation. But this is equal to ma.
Force normal to the plane = mg cosx
If M is the coefficient of friction, then , frictional force = M times normal force = M*mg cosx.
given are m = 300kg, a = 0.7 m/s^2, From scentific tables, g = 9.81m/s^2 .
So, the resultant of the force mg sinx and mg cosx is ma.Or
ma = mg sinx - M*mgcosx Or
a = g(sinx-Mcosx). Or
M cosx = (sinx - a/g). Or
M =(sinx -a/g)/cosx = (sinx - 0.7/9.981)/cosx.
M = (sinx - 0.71355759)/cosx is the coeefficient of friction between the crate and the plane.
So the value of M could be found using the different values of the adjustable inclination angle x with the help of the above equation
To get the exact numerical value of the coefficient kinetic friction we need to resolve the the force exerted by the crate in vertically downward direction in the direction parallel and perpendicular to the inclined plane. To find this we need to know the angle of the inclined plane. As this is not specified in the question it is not possible to calculate the coefficient of friction.
The angle of incline = A, and
Acceleration due to gravity = g
Then the coefficient of kinetic friction can be calculated as:
Coefficient of Kinetic friction:
= [g*Sin A - (Acceleration of crate)]/(g*CosA)
= TanA - (Acceleration of crate)/(g*CosA)
Please note that the weight of the crate is not required to calculate the coefficient of kinetic friction.