Problem 88 - College Physics

Question

\(\bullet$$\bullet\) The monkey and her bananas. \(\Lambda 20 \mathrm{kg}\) monkey has a firm hold on a light rope that passes over a frictionless pulley and is attached to a 20 \(\mathrm{kg}\) bunch of bananas (Figure 5.72 ). The monkey looks up, sees the bananas, and starts to climb the rope to get them. (a) As the monkey climbs, do the bananas move up, move down, or remain at rest? (b) As the monkey climbs, does the distance between the monkey and the ba- nanas decrease, increase, or remain con- stant? (c) The monkey releases her hold on - the rope. What happens to the distance be- tween the monkey and the bananas while she is falling? (d) Before reaching the ground, the monkey grabs the rope to stop her fall. What do the bananas do?

Step-by-Step Solution

Answer

(a) Bananas remain at rest, (b) Distance remains constant, (c) Distance remains constant during fall, (d) Bananas jerk upward.

Step 1: Analyze the Forces

The system involves the monkey, the rope, and the bananas. Since the monkey and the bananas have the same mass of 20 kg, and the pulley is frictionless, the only forces in consideration are their weights and any tension in the rope when the monkey climbs.

Step 2: Determine Movement When Monkey Climbs

When the monkey climbs the rope, she exerts a force. However, because the system is in equilibrium with equal masses on both sides, the bananas do not move. The up and down movement balances due to equal force exerted by the monkey's weight.

Step 3: Determine Relative Motion Between Monkey and Bananas

Even as the monkey climbs, the center of mass of the system does not change. Therefore, the distance between the monkey and the bananas remains constant, as climbing does not affect their relative positions.

Step 4: Analyze Fall after Monkey Releases the Rope

Upon releasing the rope, the monkey and the bananas are in free fall together. Both experience gravity equally, causing them to fall at the same rate, keeping the distance between them constant.

Step 5: Effect of Grabbing the Rope to Stop the Fall

When the monkey grabs the rope to stop her fall, it creates a sudden tension in the rope. The bananas will jerk upward momentarily due to this tension being transmitted through the rope.

Key Concepts

Newton's Laws

Newton's Laws of Motion form the foundation of classical mechanics and include three principle laws: the law of inertia, the law of acceleration, and the law of action-reaction. Each of these laws explains how objects respond to forces. In the scenario of the monkey and her bananas:

• **First Law (Inertia):** Objects at rest stay at rest, and objects in motion continue at constant speed in a straight line unless acted upon by an external force. When both the monkey and bananas are stationary, they remain so unless an external force, like the monkey climbing the rope, is applied.
• **Second Law (Acceleration):** An object’s acceleration is directly proportional to the net force acting on it and inversely proportional to its mass (\( F = ma \)). In this situation, when the monkey climbs the rope, there is no net force acting to create acceleration because the tension from both the monkey and bananas cancels each other out.
• **Third Law (Action-Reaction):** For every action, there is an equal and opposite reaction. When the monkey exerts a force climbing up, an equal force is exerted downward on the rope causing the bananas to exert force upward.

Newton's laws explain why the bananas don't move as the monkey climbs unless external forces alter the equilibrium.

Frictionless Pulley

A frictionless pulley is critical in physics problems for diminishing the complexity of the system. In an ideal scenario, like the one described:

• **No Friction Influence:** There is no friction resisting the motion of the rope over the pulley. Thus, the force applied by the monkey is entirely transferred to the bananas.
• **Smooth Motion:** The absence of friction means any motion initiated by the monkey climbing does not affect the equilibrium of forces. It helps simplify calculations, making it easier to determine outcomes based on the forces present.

Thus, in the problem, the frictionless pulley ensures that the forces applied by the monkey do not experience resistance, preserving the equal force distribution across the rope.

Equilibrium

Equilibrium in physics refers to a state where all forces acting upon an object or system are balanced, leading to a stable condition where there is no net change in motion. In the monkey and bananas scenario:

• **Equal Masses:** Both the monkey and the bananas weigh 20 kg each, contributing equally to the tension force on either side of the rope.
• **Balance of Forces:** As the monkey climbs, the tension in the rope does not change sufficiently to create an imbalance, so the bananas remain stationary. The forces are entirely balanced, maintaining equilibrium.

This balance explains why the position of the bananas does not alter when the monkey climbs; equilibrium ensures there is no net force to move the system.

Free Fall

The concept of free fall emerges when an object falls under the influence of gravity alone, without any other forces acting upon it. In the exercise:

• **Gravity's Role:** When the monkey lets go of the rope, both she and the bananas are solely under the influence of gravity. They fall together at the same acceleration, typically denoted as \(9.81 \, m/s^2\), maintaining the same distance apart.
• **Uniform Acceleration:** Free fall leads to a constant acceleration downwards for both objects, implying no differential motion between them. This explains why the gap between the monkey and the bananas stays unchanged during the fall.

Understanding free fall helps illustrate why objects in a system like this one, without air resistance or other forces, behave predictably under Earth's gravitational pull.

Tension Forces

Tension forces arise in a rope or cable when a force is applied to either end, often caused by pulling actions. In the monkey and bananas problem, tension plays a central role:

• **Equal Tension Distribution:** As both the monkey and bananas exert force, the tension remains uniform throughout the rope due to the equal weights on either side.
• **Impact of Actions:** When the monkey climbs, she changes the internal forces momentarily but the equal weights keep the system in balance. Tension is the reason why the bananas do not shift, even with her movements.
• **Tension Upon Halt:** When the monkey grabs the rope again to stop falling, the tension suddenly increases, causing the bananas to jerk upward momentarily.

Tension means that forces are transmitted but remain balanced due to the equal weights, which is why understanding tension is crucial for solving pulley problems.