A small block placed on the surface of a horizontal disc at a point 6 cm from the centre of the disc .The block is on the point of slipping when the disc rotates at 1 1/2 rev/s .Find the coefficient of friction between the block and the surface of the disc.
To find the coefficient of friction between the block and the surface of the disc, we need to use the condition for slipping.
When the block is on the point of slipping, the maximum static friction force between the block and the surface of the disc is equal to the centripetal force acting on the block.
The maximum static friction force, Fmax, can be calculated as:
Fmax = μN
where μ is the coefficient of friction and N is the normal force.
The normal force, N, can be calculated as:
N = mg
where m is the mass of the block and g is the acceleration due to gravity.
The centripetal force, Fc, can be calculated as:
Fc = mrω²
where r is the distance of the block from the center of the disc, ω is the angular velocity of the disc, and m is the mass of the block.
We are given that the distance of the block from the center of the disc, r, is 6 cm. We also know that the angular velocity of the disc, ω, is 1 1/2 rev/s, or (3/2) × 2π rad/s.
Now, let's substitute these values into the equations to find the coefficient of friction, μ.
N = mg = m × 9.8 m/s²
Fc = mrω² = m × (6/100) m × (3/2) × 2π rad/s²
Since Fc = Fmax, we can equate these equations:
m × 9.8 m/s² = m × (6/100) m × (3/2) × 2π rad/s²
Simplifying the equation:
9.8 m/s² = (6/100) × (3/2) × 2π rad/s²
m cancels out from both sides, and we are left with:
9.8 = (6/100) × (3/2) × 2π rad/s²
Simplifying further:
9.8 = (6/100) × (3/2) × 2π rad/s²
9.8 = (6/100) × (3/2) × 2π rad/s²
9.8 = (18/100) × π rad/s²
Dividing both sides by π:
9.8/π = 18/100 rad/s²
Simplifying:
0.3125 = 18/100 rad/s²
Now, we can isolate the coefficient of friction, μ:
Fmax = μN
μ = Fmax/N
Substituting the values, we get:
μ = (9.8/π)/(m × 9.8 m/s²)
μ = 1/π
Therefore, the coefficient of friction between the block and the surface of the disc is 1/π, or approximately 0.318.
To find the coefficient of friction between the block and the surface of the disc, we can use the concept of circular motion and centripetal force.
Let's denote the coefficient of friction as μ.
The centripetal force required to keep the block in circular motion is provided by the frictional force acting on the block. This force can be given by:
Frictional force = Centripetal force
The centripetal force can be calculated using the following formula:
Centripetal force = (mass of the block) x (linear velocity squared) / (radius)
Given that the block is on the point of slipping, the frictional force will be at its maximum (μN), where N is the normal force acting on the block due to the weight.
The normal force (N) can be calculated using:
N = (mass of the block) x (acceleration due to gravity)
Linear velocity can be calculated using the formula:
Linear velocity = (angular velocity) x (radius)
Given that the disc rotates at 1 1/2 rev/s, the angular velocity can be calculated as:
Angular velocity = (1 1/2 rev/s) x (2π rad/rev)
Once we have all these values, we can equate the frictional force to the centripetal force and solve for the coefficient of friction (μ).
Let's calculate step-by-step:
1. Convert the angular velocity to rad/s:
angular velocity = (1.5 rev/s) x (2π rad/rev) = 3π rad/s
2. Convert the radius from cm to m:
radius = 6 cm = 0.06 m
3. Calculate the linear velocity:
linear velocity = angular velocity x radius
= (3π rad/s) x (0.06 m)
4. Calculate the normal force:
mass of the block = m (given)
acceleration due to gravity = g ≈ 9.8 m/s^2
N = m x g
5. Calculate the centripetal force:
centripetal force = (mass of the block) x (linear velocity^2) / (radius)
= (m) x [(angular velocity x radius)^2] / (radius)
= m x (angular velocity^2) x radius
6. Equate the frictional force to the centripetal force:
μN = centripetal force
Substitute the values of N and centripetal force:
μ(m x g) = m x (angular velocity^2) x radius
7. Simplify the equation:
μg = (angular velocity^2) x radius
8. Solve for the coefficient of friction (μ):
μ = ((angular velocity^2) x radius) / g
Substitute the values of angular velocity, radius, and g:
μ = [(3π rad/s)^2 x (0.06 m)] / (9.8 m/s^2)
Evaluate the expression to find the coefficient of friction (μ).
By following these steps and performing the calculations, you can find the coefficient of friction between the block and the surface of the disc.