A wooden block of mass 9 kg is at rest on an inclined plane sloped at an angle theta from the horizontal. The block is located 5 m from the bottom of the plane.
A) Find the frictional force for theta = 20
B)If the angle is increased slowly the block starts sliding at an angle theta = 30. What is the coefficient of static friction?
C) If the block slides to the bottom in 2 s, what is the coefficient of kinetic friction?
M*g = 9 * 9.8 = 88.2 N. = Wt. of block.
A. Fp = 88.2*sn20= 30.2 N. = Force
parallel to the in.cline.
Fn = 88.2*Cos20 = 82.9 N. = Normal or force perpendicular to the incline.
Fs = u*Fn = u*82.9.
Fp-Fs = M*a. 30.2-82.9u = M*0 = 0, 82.9u = 30.2, u = 0.364 = Static coefficient of friction.
Fs = u*Fn = 0.364 * 82.9 = 30.2 N. = Force of static friction.
B. Fp = 88.2*sin30 = 44.1 N.
Fn = 88.2*Cos30 = 76.4 N.
Fs = u*Fn = u*76.4.
Fp-Fs = M*a.
44.1 - 76 .4u = M*0 = 0, u = 0.577.
A) To find the frictional force for theta = 20, we need to consider the forces acting on the wooden block. In this case, we have the force due to gravity, the normal force, and the frictional force.
1. The force due to gravity (Fg) can be calculated using the formula Fg = m * g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
For a wooden block with a mass of 9 kg, the force due to gravity would be:
Fg = 9 kg * 9.8 m/s^2 = 88.2 N
2. The normal force (Fn) is the force exerted perpendicular to the inclined plane and can be calculated using the formula Fn = m * g * cos(theta), where theta is the angle of the inclined plane.
For theta = 20, the normal force would be:
Fn = 9 kg * 9.8 m/s^2 * cos(20°)
3. The frictional force (Ff) can be calculated using the formula Ff = f * Fn, where f is the coefficient of friction.
To find the frictional force, we need to know the coefficient of friction between the block and the inclined plane. Given that the block is at rest, we can assume this is the coefficient of static friction (fs).
B) If the block starts sliding at an angle theta = 30, it means that the force of static friction has been overcome.
We can find the coefficient of static friction using the following formula: fs = tan(theta)
For theta = 30°,
fs = tan(30°)
C) To find the coefficient of kinetic friction, we will use the formula for acceleration on an inclined plane. When the block is sliding, the frictional force becomes the kinetic friction (fk).
1. The acceleration (a) of the block is given by the formula a = g * sin(theta).
Given that the block slides to the bottom in 2 seconds, we know that the distance covered (s) is 5 m. We can use the following kinematic equation to find the acceleration:
s = ut + (1/2) * a * t^2,
where u is the initial velocity (which is 0 m/s) and t is the time taken (2 s).
Rearranging the equation, we get:
a = 2s / t^2
2. The frictional force (Ff) is given by the formula Ff = m * g * sin(theta) - fk * m * g * cos(theta).
Since the block slides to the bottom, we can assume that the frictional force Ff is equal to the force due to gravity Fg.
Setting Ff = Fg and rearranging the equation, we get:
fk = (m * g * sin(theta)) / (m * g * cos(theta))
Simplifying, we find:
fk = tan(theta),
where theta is the angle at which the block starts sliding.