An object of mass 20kg slides down an inclined plane at an angle of 30 degree to the horizontal. The coefficient of static friction is?
If 30 degrees is the minimum angle that allows slipping to start, then the coefficient of static friction (mu_s) is given by
M*g*sin30 = M*g*mu_s*cos30
mu_s = tan30 = 0.5774
The mass M does not matter.
Angles greater than 30 degrees will also allow slipping.
0.5774
What is the meaning of mus
To find the coefficient of static friction in this scenario, we need to use the given information about the object's mass, the angle of the inclined plane, and the fact that the object is sliding.
First, let's define the forces acting on the object as it slides down the inclined plane. There are two main forces at play: the component of gravitational force that pulls the object down the slope, and the force of friction that opposes the object's motion.
1. Gravitational Force:
The component of the gravitational force parallel to the inclined plane is given by:
F_parallel = m * g * sin(θ)
where
m = mass of the object (20 kg)
g = acceleration due to gravity (approximately 9.8 m/s^2)
θ = angle of incline (30 degrees)
2. Force of Friction:
The force of friction opposes the motion of the object. So, when the object is sliding down, the force of friction is kinetic friction. The formula for kinetic friction is:
F_friction = μ * N
where
μ = coefficient of kinetic friction
N = normal force
To find the normal force, we need to consider the vertical component of the gravitational force:
F_vertical = m * g * cos(θ)
Since there is no vertical acceleration and the object is not moving vertically, the normal force is equal to the vertical component of the gravitational force:
N = m * g * cos(θ)
Since the object is sliding, the force of kinetic friction must be equal to the gravitational force parallel to the incline:
F_friction = F_parallel
Now, we can substitute the known values into the equations to solve for the coefficient of kinetic friction:
μ * N = m * g * sin(θ)
μ * (m * g * cos(θ)) = m * g * sin(θ)
Simplifying the equation:
μ * cos(θ) = sin(θ)
μ = sin(θ) / cos(θ)
Now, we can substitute the angle of the incline into the equation to find the coefficient of kinetic friction:
μ = sin(30°) / cos(30°)
Using trigonometric identities:
μ = (1/2) / (√3/2)
μ = 1 / √3
μ ≈ 0.577
Hence, the coefficient of static friction is approximately 0.577 (or 1/√3).