Question

A pulley is hung from the ceiling by a rope. A block of mass \(M\) is suspended by another rope that passes over the pulley and is attached to the wall. The rope fastened to the wall makes a right angle with the wall. Ignore the masses of the rope and the pulley. Find (a) the tension in the rope from which the pulley hangs and (b) the angle \(\theta\) that the rope makes with the ceiling.

Short answer

(a) The tension is \(2Mg\). (b) The angle \(\theta\) is \(60^\circ\).

Step-by-step solution

Understanding the Problem

We have a pulley system where a block of mass \(M\) is hung and connected through a rope over a pulley to the wall. We are tasked with finding two things: (a) the tension in the rope from which the pulley hangs, and (b) the angle \(\theta\) that this rope makes with the ceiling.

Analyze Forces on the Block

Since the block is in equilibrium, the tension \(T\) in the rope supporting the block must equal the gravitational force acting on the block. Thus, we have: \[ T = Mg \] where \(g\) is the acceleration due to gravity.

Determine Forces on the Pulley

The pulley is also in equilibrium. Two tensions are acting vertically downward due to the rope from both sides (around the block and towards the wall): each of magnitude \(T = Mg\). This creates a net downward force.

Find Tension in the Hanging Rope (Part a)

The tension in the rope from which the pulley hangs, let's call it \(T_{p}\), must balance the vertical forces due to the two downward tensions. Thus, \[ T_{p} = 2T = 2Mg \].

Understanding the Geometry of the Ropes (Part b)

The rope towards the wall makes a right angle with the wall, making the setup form a triangle with one angle being \(90^\circ\). The horizontal component of \(T\) will affect the tension in the hanging rope.

Calculate the Angle \(\theta\) (Part b)

Consider the equilibrium in both horizontal and vertical directions. Horizontally, tension \(T\) equals the horizontal component of \(T_{p}\): \[ T_{p}\cos(\theta) = T \].Substitute \(T = Mg\) and \(T_{p} = 2Mg\), giving us:\[ 2Mg \cdot \cos(\theta) = Mg \], solving for \(\cos(\theta)\), we get \(\cos(\theta) = \frac{1}{2}\).Thus, \(\theta = 60^\circ\).

Key concepts

Tension in Ropes

When studying pulley systems, understanding tension in the ropes is crucial. Tension is essentially the force exerted along the rope. In the case of our pulley system, the block's mass creates a downward gravitational force, which translates into tension along the rope. This tension helps maintain the block's equilibrium. Since the block weighs down with a force of gravity (Mg), the tension in the rope must counterbalance this. This means the force, or tension, is equal to the weight of the block:

• The formula for tension here is: \[ T = Mg \]

The tension is the same throughout the rope, a characteristic property of ideal ropes in physics problems where the rope's mass and any friction are neglected. This uniform tension supports pulling or lifting objects while distributing force evenly across the system. It is crucial to remember that in reality, factors like rope mass and friction can affect tension slightly, but for ideal systems, we focus on the simplified model.

Equilibrium in Physics

Equilibrium is a fundamental concept in physics. It describes a state where all forces acting on a system balance each other, resulting in a stable condition with no net force. In our exercise, both the block and the pulley are in equilibrium, meaning there’s no acceleration involved—the system remains at rest or moves with constant velocity. The tension in that system holds the block stationary, counteracting gravity effectively.For equilibrium:

• Net force in any direction is zero.
• For vertical direction: \[ ext{Net force} = T - Mg = 0 \] This implies that the downward gravitational force equals the upward tension.
• For the pulley, multiple tensions exert forces to keep it stable, showing equilibrium.

Understanding equilibrium helps determine how forces interact in physical systems, ensuring stability, whether for stationary objects or objects in uniform motion.

Trigonometry in Physics

Trigonometry is a useful tool in physics, especially when dealing with forces at angles, such as those in our pulley system problem. The hanging rope forms an angle with the ceiling, affecting the tension's horizontal and vertical components. By using trigonometric principles, you can resolve these components and find the precise angle needed.Consider the trigonometric components:

• Horizontal Component: It involves the tension due to the wall-attached rope, calculated using: \[ T_{horizontal} = T_{p} imes ext{cos}(\theta) \]
• Vertical Component: Here, the total downward tension should balance with the upward tension of the hanging rope:

To find the angle \( \theta \), we calculate using the relation where the horizontal component of tension in the hanging rope equals the tension on the block's side. Given:

• \( T_{p} \cdot \text{cos}(\theta) = T \) leads to \( \text{cos}(\theta) = \frac{1}{2} \)
• This trigonometric relationship indicates that the angle is \( \theta = 60^{\circ} \).

Trigonometry allows us to break down complex force interactions into simpler components, aiding in solving problems involving forces at angles. It’s a powerful tool for understanding and solving physics problems.