Question

A $10 \mathrm{~kg}$ monkey climbs up a massless rope that runs over a frictionless tree limb and back down to a $15 \mathrm{~kg}$ package on the ground (Fig. 5-44). (a) What is the magnitude of the least acceleration the monkey must have if it is to lift the package off the ground? If, after the package has been lifted, the monkey stops has becn lifted, the monkey stops its climb and holds on to the rope, what are the (b) magnitude and (c) what are the (b) magnitude and (c) direction of the monkey's acceleration and (d) the tension in the rope? 

    A $10 \mathrm{~kg}$ monkey climbs up a massless rope that runs over a frictionless tree limb and back down to a $15 \mathrm{~kg}$ package on the ground (Fig. 5-44). (a) What is the magnitude of the least acceleration the monkey must have if it is to lift the package off the ground? If, after the package has been lifted, the monkey stops has becn lifted, the monkey stops its climb and holds on to the rope, what are the (b) magnitude and (c) what are the (b) magnitude and (c) direction of the monkey's acceleration and (d) the tension in the rope?
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Principles of Physics
Principles of Physics
David Halliday ,… 10th Edition
Chapter 5, Problem 59 ↓

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The tension in the rope is denoted as $T$. The force equation for the monkey is $T - M_1g = M_1a_{min}$, where $g$ is the acceleration due to gravity and $a_{min}$ is the minimum acceleration the monkey must have to lift the package off the ground.  Show more…

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A $10 \mathrm{~kg}$ monkey climbs up a massless rope that runs over a frictionless tree limb and back down to a $15 \mathrm{~kg}$ package on the ground (Fig. 5-44). (a) What is the magnitude of the least acceleration the monkey must have if it is to lift the package off the ground? If, after the package has been lifted, the monkey stops has becn lifted, the monkey stops its climb and holds on to the rope, what are the (b) magnitude and (c) what are the (b) magnitude and (c) direction of the monkey's acceleration and (d) the tension in the rope?
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Key Concepts

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Newton's Second Law
Newton's Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. It is fundamental in dynamics and is used to determine the acceleration of individual bodies when the forces acting on them are known, as well as analyzing the net forces in system problems.
Free-Body Diagrams
Free-body diagrams are graphical representations that depict the forces acting on each object within a system. They allow for the systematic identification of all forces, such as gravitational forces and tensions, making it easier to apply Newton's Second Law and solve for unknown quantities.
Massless and Frictionless Rope and Pulley Systems
In systems involving massless ropes and frictionless pulleys, the tension is the same throughout the rope. This idealization simplifies the analysis as it means that the force transmitted through the rope does not change between different parts of the system, and rotational inertia of pulleys can be ignored.
Constraint Relations in Pulley Systems
Pulley systems impose kinematic constraints that relate the accelerations of different masses connected by the rope. These relations arise from the fixed length of the rope, ensuring that any acceleration of one mass must be accompanied by a corresponding acceleration of the other, albeit possibly in opposite directions.
Minimum Acceleration to Overcome Weight Differences
When two masses are connected in a system, the heavier mass can act as a counterweight. Determining the minimum acceleration needed to lift one mass involves calculating the net force required to overcome the gravitational force imbalance, ensuring that the system starts moving in the desired direction.
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A 10 $\mathrm{kg}$ monkey climbs up a massless rope that runs over a frictionless tree limb and back down to a 15 $\mathrm{kg}$ package on the ground (Fig. $5-54 )$ . (a) What is the magnitude of the least acceleration the monkey must have if it is to lift the package off the ground? If, after the package has been lifted, the monkey stops its climb and holds onto the rope, what are the (b) magnitude and (c) direction of the monkey's acceleration and (d) the tension in the rope?

Fundamentals of Physics

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A 10 $\mathrm{kg}$ monkey climbs up a massless rope that runs over a frictionless tree limb and back down to a 15 $\mathrm{kg}$ package on the ground (Fig. $5-54 )$ . (a) What is the magnitude of the least acceleration the monkey must have if it is to lift the package off the ground? If, after the package has been lifted, the monkey stops its climb and holds onto the rope, what are the (b) magnitude and (c) direction of the monkey's acceleration and (d) the tension in the rope?
a-10-mathrmkg-monkey-climbs-up-a-massless-rope-that-runs-over-a-frictionless-tree-limb-and-back-down-to-a-15-mathrmkg-package-on-the-ground-fig-5-54-a-what-is-the-magnitude-of-the-least-acceleration-t

A 10 $\mathrm{kg}$ monkey climbs up a massless rope that runs over a frictionless tree limb and back down to a 15 $\mathrm{kg}$ package on the ground (Fig. $5-54 )$ . (a) What is the magnitude of the least acceleration the monkey must have if it is to lift the package off the ground? If, after the package has been lifted, the monkey stops its climb and holds onto the rope, what are the (b) magnitude and (c) direction of the monkey's acceleration and (d) the tension in the rope?
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