A long non-stretchable rope is thrown over a stationary block. At one end of the rope hangs a weight of mass $m = 10 kg$, the other end is grasped by a monkey of mass $M = 30 kg$. With what acceleration relative to the rope must the monkey move to remain at the same height above the Earth's surface? Neglect the mass of the rope and the friction in the block.
Decision:
Due to the weightlessness of the rope, the tension force $T$ is the same in any section of the rope. The forces acting on the monkey are: rope tension force $T$ - upward, gravity force $Mg$ - downward. Under the action of these forces, the monkey must rest relative to the earth, i.e. its equation of motion according to Newton's law of motion
$T - Mg = 0$. (1)
The forces acting on the weight are $T$ upward and $mg$ downward, causing the weight to move upward relative to the ground with acceleration $a$:
$T - mg = ma$. (2)
Solving the system of equations (1), (2), we find the acceleration:
$a=g \left ( \frac{M}{n}-1 \right )$
The rope's inextensibility means that it moves relative to the ground with the same acceleration $a = 2g$. Consequently, the acceleration of the rope and relative to the monkey is $2g$, from which we conclude that the monkey must move relative to the rope with an acceleration of $2g$.
Decision:
Due to the weightlessness of the rope, the tension force $T$ is the same in any section of the rope. The forces acting on the monkey are: rope tension force $T$ - upward, gravity force $Mg$ - downward. Under the action of these forces, the monkey must rest relative to the earth, i.e. its equation of motion according to Newton's law of motion
$T - Mg = 0$. (1)
The forces acting on the weight are $T$ upward and $mg$ downward, causing the weight to move upward relative to the ground with acceleration $a$:
$T - mg = ma$. (2)
Solving the system of equations (1), (2), we find the acceleration:
$a=g \left ( \frac{M}{n}-1 \right )$
The rope's inextensibility means that it moves relative to the ground with the same acceleration $a = 2g$. Consequently, the acceleration of the rope and relative to the monkey is $2g$, from which we conclude that the monkey must move relative to the rope with an acceleration of $2g$.