First slide

Rolling motion

Question

A sphere of radius r and mass m has a linear velocity v₀ m/s directed to the left and no angular velocity as it is placed on a horizontal platform moving to the right with
a constant velocity 10 m/s. If after sliding on the platform the sphere is to have no linear velocity relative to the ground as it starts rolling on the platform without sliding.
The coefficient of kinetic friction between the sphere and the platform is $μ_{k}$ . Determine the required value of v₀ in m/s.

Difficult

Solution

$\begin{array}{l} A n g u l a r I m p u l s e - A n g u l a r M o m e n t u m : f R Δ t = I_{cm} ω \\ L i n e a r I m p u l s e - L i n e a r M o m e n t u m : - f Δ t = m (0 - v_{0}) \\ P u r e R o l l i n g : ω r = v_{1} \\ \Rightarrow m v_{0} R = \frac{2}{5} m R^{2} ω \\ \Rightarrow ω = \frac{5 v_{0}}{2 R} \\ \Rightarrow v_{1} = \frac{5 v_{0}}{2} \\ \Rightarrow v_{0} = \frac{2 v_{1}}{5} \\ \Rightarrow v_{0} = \frac{2 \times 10}{5} = 4 m / s \end{array}$

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