A hoop of mass 'm' is projected on a floor with linear velocity 'v' and reverse spin 'w'. The
co efficient of friction is 'u'. What is the
(i) time of pure rolling?
(ii) velocity of return?
To find the time of pure rolling and the velocity of return for a hoop rolling on a floor, we'll need to consider the forces acting on the hoop and use the concept of rotational motion. Here's how we can calculate these quantities:
(i) Time of Pure Rolling:
During pure rolling, the hoop moves without any slipping. This means that the linear velocity of the center of the hoop is related to its angular velocity by the equation v = R * ω, where R is the radius of the hoop.
To find the time of pure rolling, we need to determine how long it takes for the hoop to come to rest due to friction. The frictional force acting on the hoop opposes its motion and can be calculated using the equation f_friction = u * m * g, where u is the coefficient of friction, m is the mass of the hoop, and g is the acceleration due to gravity.
The frictional force can also be expressed as the torque applied to the hoop. The torque caused by friction can be calculated as τ = f_friction * R, where R is the radius of the hoop.
The torque due to friction causes negative angular acceleration, given by α = τ / I, where I is the moment of inertia of the hoop.
For a hoop, the moment of inertia is I = m * R^2.
Using the equations of rotational motion, we can relate the angular acceleration to the angular velocity and time of pure rolling:
ω = ω_0 + α * t, where ω_0 is the initial angular velocity of the hoop.
Since the hoop starts with reverse spin, ω_0 = -w.
To find the time of pure rolling, we can set ω = 0 and solve for t:
0 = -w + (-τ / I) * t
Substituting the values, we get:
0 = -w + [-(u * m * g * R) / (m * R^2)] * t
Simplifying, we have:
w * R^2 = u * g * t
Finally, solving for t, the time of pure rolling:
t = (w * R^2) / (u * g)
(ii) Velocity of Return:
The velocity of return can be calculated by considering the motion of the center of the hoop. Since the hoop is moving with linear velocity v and rolling without slipping, the velocity of return can be calculated as:
v_return = -v
The negative sign indicates that the direction of the return velocity is opposite to the initial linear velocity.
Therefore, the velocity of return is simply the negative of the initial linear velocity.
Note: This explanation assumes that the hoop is rolling on a flat surface without any external forces acting on it.