Question

Two masses \(\mathrm{M}\) and \((\mathrm{M} / 2)\) are joined together by means of light inextensible string passed over a frictionless pulley as shown in fig. When the bigger mass is released, the small one will ascend with an acceleration (A) \((\mathrm{g} / 3)\) (B) \((3 \mathrm{~g} / 2)\) (C) \(\mathrm{g}\) (D) \((\mathrm{g} / 2)\)

Short answer

The acceleration of the smaller mass is \(a = \frac{g}{3}\), so the correct answer is (A) \(\frac{g}{3}\).

Step-by-step solution

Define the forces acting on each mass

First, let's identify and label the forces acting on each mass. For mass M, there is a downward gravitational force, which can be expressed as \(F_{Mg} = M \cdot g\), where g is the acceleration due to gravity. Similarly, for mass M/2, there is a downward gravitational force \(F_{(M/2)g} = (M/2) \cdot g\). There is also tension in the string, which we will denote as T. The tension acts upward on both masses.

Write the equations of motion for each mass

According to Newton's second law of motion, the net force acting on an object is equal to its mass multiplied by its acceleration. Therefore, we can write the equations of motion for both masses: 1. For mass M: \(M \cdot a = M \cdot g - T\) 2. For mass M/2: \((M/2) \cdot a = T - (M/2) \cdot g\) Now we have a system of two equations with two unknowns (acceleration a and tension T).

Solve the system of equations

We will first solve the equation for mass M to express the tension T in terms of acceleration a: \(T = M \cdot g - M \cdot a\) Now, substitute this expression for tension T in the equation for mass M/2: \((M/2) \cdot a = (M \cdot g - M \cdot a) - (M/2) \cdot g\)

Solve for acceleration a

Now, we can solve the equation for acceleration a: \((M/2) \cdot a + M \cdot a = M \cdot g - (M/2) \cdot g\) \((3/2)Ma = (1/2)Mg\) Divide both sides by \(3/2M\): \(a = \frac{1/2 \cdot g}{3/2} = \frac{g}{3}\) The acceleration of the smaller mass is \(a = \frac{g}{3}\). So the correct answer is (A) \(\frac{g}{3}\).

Key concepts

Pulley System

In our exercise, the pulley system is a fundamental setup used to explore forces and motions. A pulley system comprises one or more pulleys and a rope or string, which is utilized to move objects up and down or across spaces. The beauty of this system lies in its simplicity and efficiency.
This system helps to change the direction of applied force, making it easier to lift heavy objects with less effort. In an ideal situation, like the one described, the pulley is frictionless. This assumption allows us to concentrate on the forces acting on the masses without worrying about additional resistance that would complicate our calculations.

• The pulley changes the direction of the tension force in the string.
• In this scenario, when one mass is released, the opposing mass moves in the opposite direction.

This symmetrical movement is determined by the difference in mass and tension, with the heavier mass traveling downwards, causing the lighter one to ascend.

Tension in Strings

Tension in the string is another essential concept to grasp. Tension acts along the length of the string and is the result of forces pulling at it from either end. It is crucial in determining how the system behaves.
For both masses in our pulley problem, the tension is the same throughout the string because it is light and inextensible. This uniformity ensures that the tension force acting upward on one mass is equal to the tension acting downward on the other mass.

• Tension opposes gravitational forces acting on the masses.
• It provides the necessary counteractive force that allows us to determine the net force and subsequently the acceleration of the system.

The equations of motion stemmed from understanding that the tension combats the gravitational pull on each mass. Solving these equations enables us to find the tension and, importantly, the acceleration that dictates the system's behavior.

Acceleration Due to Gravity

Acceleration due to gravity, denoted by \(g\), is a crucial constant when dealing with earthly physical problems. It typically holds a value of approximately \(9.8 \/\text{m/s}^2\) and signifies how fast the velocity of an object changes due to gravitational pull.
In our pulley exercise, gravity acts on both masses, pulling them downward. The task is to determine how this pull interacts with the pulley system to cause the masses to accelerate.

• Gravity is the only external force causing the system to move since the pulley is frictionless.
• The different masses create an imbalance, which results in motion—showing Newton's Laws of Motion in action.

When resolving the exercise, the gravitational force provides the baseline from which we calculate net forces and, inevitably, the acceleration of the mass system. The acceleration we determined, \(a = \frac{g}{3}\), is a fraction of \(g\), indicating that the string and pulley effectively slow down the fall of the heavier weight and the rise of the lighter one.