A weightless rope is thrown over a log. A body of mass $m$ is tied to one end of the rope, and a force $F$ directed vertically downward is applied to the other end. Because of the friction between the rope and the log, the minimum value of the force at which the body $m$ hangs motionless is $\alpha mg$, where $\alpha$ is less than unity and independent of the unit $m$. Find the minimum value of the force $F$ at which body $m$ will rise upward.
Decision:
When the force $F$ changes from $F_{min}=\alpha mg$ to $F_{max}$, body $m$ is stationary. A slight increase in force $F$ over $F_{max}$ will cause the body $m$ to rise. Hence, $F_{max}$ is the force to be found according to the problem condition.
So, according to the problem condition, the rope does not move if a force $mg$ is applied to one end of the rope and $F = \alpha mg$ is applied to the other end. Of particular importance in the problem condition is the statement that $ \alpha$ does not depend on the mass, and hence on the magnitude of the force $mg$. If an arbitrary force $F^{\prime}$ is applied to one end of the rope, then it is sufficient to apply a force $ \alpha F^{\prime} < F^{\prime} < F^{\prime}$ to the other end to keep the rope stationary. Suppose a force $F^{\prime}= mg/ \alpha$ is applied to one end of the rope and $mg < F^{\prime}$ is applied to the other end . The rope is stationary. When the force $F^{\prime}$ is slightly increased, the body $m$ starts to rise upwards.
Thus, the required minimum value of the force at which the body of mass $m$ rises up is $mg/ \alpha$
Decision:
When the force $F$ changes from $F_{min}=\alpha mg$ to $F_{max}$, body $m$ is stationary. A slight increase in force $F$ over $F_{max}$ will cause the body $m$ to rise. Hence, $F_{max}$ is the force to be found according to the problem condition.
So, according to the problem condition, the rope does not move if a force $mg$ is applied to one end of the rope and $F = \alpha mg$ is applied to the other end. Of particular importance in the problem condition is the statement that $ \alpha$ does not depend on the mass, and hence on the magnitude of the force $mg$. If an arbitrary force $F^{\prime}$ is applied to one end of the rope, then it is sufficient to apply a force $ \alpha F^{\prime} < F^{\prime} < F^{\prime}$ to the other end to keep the rope stationary. Suppose a force $F^{\prime}= mg/ \alpha$ is applied to one end of the rope and $mg < F^{\prime}$ is applied to the other end . The rope is stationary. When the force $F^{\prime}$ is slightly increased, the body $m$ starts to rise upwards.
Thus, the required minimum value of the force at which the body of mass $m$ rises up is $mg/ \alpha$