To solve the problem of a solid sphere sliding on a plank and transitioning to pure rolling, we need to analyze the forces and motions involved. The key here is to understand how friction affects the sphere's motion and how it leads to rolling without slipping.
Understanding the Forces at Play
When the sphere is given a sharp impulse, it starts sliding with an initial speed \( v \). The frictional force between the sphere and the plank will act to oppose this sliding motion. This frictional force is what will eventually cause the sphere to start rolling.
Key Variables
- m: mass of the sphere (and the plank)
- R: radius of the sphere
- u: coefficient of friction between the sphere and the plank
- f: frictional force acting on the sphere
Frictional Force and Acceleration
The maximum static frictional force that can act on the sphere is given by:
f = u \cdot m \cdot g
where \( g \) is the acceleration due to gravity. This frictional force will cause the sphere to decelerate while also providing a torque that will lead to rolling.
Equations of Motion
When the sphere starts sliding, it experiences a linear deceleration due to friction:
a = -\frac{f}{m} = -\frac{u \cdot m \cdot g}{m} = -u \cdot g
Simultaneously, the torque \( \tau \) caused by the frictional force about the center of the sphere is:
\tau = f \cdot R = u \cdot m \cdot g \cdot R
This torque results in an angular acceleration \( \alpha \) given by:
\alpha = \frac{\tau}{I} = \frac{u \cdot m \cdot g \cdot R}{\frac{2}{5} m R^2} = \frac{5u \cdot g}{2R}
Here, \( I = \frac{2}{5} m R^2 \) is the moment of inertia of the sphere.
Time to Transition to Pure Rolling
To find the time taken for the sphere to transition to pure rolling, we need to determine when the linear velocity \( v_f \) of the sphere equals the angular velocity \( \omega \) times the radius \( R \). The relationship is:
v_f = \omega R
Initially, the sphere has a linear velocity \( v \) and no angular velocity. As time progresses, the linear velocity decreases due to the frictional force, and the angular velocity increases due to the torque:
v_f = v - u \cdot g \cdot t
\omega = \alpha \cdot t = \frac{5u \cdot g}{2R} \cdot t
Setting Up the Equation
At the point of pure rolling, we set the two expressions equal:
v - u \cdot g \cdot t = \left(\frac{5u \cdot g}{2R} \cdot t\right) R
Rearranging gives us:
v = u \cdot g \cdot t + \frac{5u \cdot g}{2} \cdot t
Combining terms leads to:
v = \left(u \cdot g + \frac{5u \cdot g}{2}\right) t = \frac{7u \cdot g}{2} t
Final Calculation
Now, we can solve for \( t \):
t = \frac{2v}{7u \cdot g}
This equation gives us the time taken for the sphere to transition from sliding to pure rolling on the plank.
In summary, by analyzing the forces and motions involved, we derived the time it takes for the sphere to start rolling without slipping. This approach not only highlights the importance of friction but also illustrates the interplay between linear and angular motion in rotational dynamics.
