Question

In the arrangement shown in figure $a_1, a_2, a_3$ and $a_4$ are the accelerations of masses $m_1, m_2, m_3$ and $m_4$ respectively. Which of the following relation is true for this arrangement?

Answer 1

a_1 + 2a_2 = 0

Answer 2

To determine the relation between the accelerations of the masses, we use the constraint due to the inextensibility of the string.

Let's define a coordinate system. Let the fixed ceiling be the origin ($y=0$). We will define downward as the positive direction for positions and accelerations.

1. Positions of the masses and pulleys:

   • Let $y_1$ be the position of mass $m_1$ from the ceiling.
   • Let $y_P$ be the position of the movable pulley from the ceiling.
   • Mass $m_2$ is attached directly to the movable pulley, so its position $y_2$ is the same as the position of the movable pulley, i.e., $y_2 = y_P$.

2. Length of the string: The string passes over the fixed pulley and then around the movable pulley. One end of the string is attached to mass $m_1$, and the other end is attached to the ceiling.

   • The segment of the string connecting mass $m_1$ to the fixed pulley has a length proportional to $y_1$. Let's assume the fixed pulley is at $y=0$. Then this length is $y_1$.
   • The segment of the string connecting the fixed pulley to the movable pulley has a length proportional to $y_P$. This length is $y_P$.
   • The segment of the string connecting the movable pulley to the ceiling attachment point also has a length proportional to $y_P$. This length is also $y_P$.
   • There are also constant lengths of the string that are in contact with the fixed and movable pulleys. Let this constant length be $C$.

   The total length of the string, $L$, is constant: $L = y_1 + y_P + y_P + C$ $L = y_1 + 2y_P + C$

3. Differentiating to find velocities and accelerations: Since the string is inextensible, its total length $L$ is constant. Differentiating the equation for $L$ with respect to time ($t$): $\frac{dL}{dt} = \frac{dy_1}{dt} + 2\frac{dy_P}{dt} + \frac{dC}{dt}$ Since $L$ and $C$ are constants, $\frac{dL}{dt} = 0$ and $\frac{dC}{dt} = 0$. So, $0 = v_1 + 2v_P$, where $v_1 = \frac{dy_1}{dt}$ is the velocity of $m_1$, and $v_P = \frac{dy_P}{dt}$ is the velocity of the movable pulley.

   Differentiating again with respect to time to find accelerations: $\frac{d(0)}{dt} = \frac{dv_1}{dt} + 2\frac{dv_P}{dt}$ $0 = a_1 + 2a_P$, where $a_1 = \frac{dv_1}{dt}$ is the acceleration of $m_1$, and $a_P = \frac{dv_P}{dt}$ is the acceleration of the movable pulley.

4. Relating pulley acceleration to mass acceleration: Since mass $m_2$ is attached directly to the movable pulley, its acceleration $a_2$ is the same as the acceleration of the movable pulley: $a_2 = a_P$

5. Final relation: Substitute $a_P = a_2$ into the equation from step 3: $a_1 + 2a_2 = 0$

This relation indicates that if $m_1$ accelerates downwards (positive $a_1$), then $m_2$ accelerates upwards (negative $a_2$), and the magnitude of $a_1$ is twice the magnitude of $a_2$.

The question mentions $a_3$ and $a_4$ for masses $m_3$ and $m_4$, but the diagram only shows $m_1$ and $m_2$. It is highly probable that the mention of $m_3$ and $m_4$ is a typo or refers to an unpictured part of a larger system, and the intended question is to find the relation between $a_1$ and $a_2$ for the given setup.

The final answer is $a_1 + 2a_2 = 0$.

Explanation of the Solution:

The system consists of an inextensible string passing over a fixed pulley and a movable pulley. By defining positions from a fixed reference (e.g., the ceiling) and using the constant length of the string, a kinematic constraint equation is derived. Let $y_1$ be the position of $m_1$ and $y_P$ be the position of the movable pulley (which is the same as $y_2$ for $m_2$). The string length $L = y_1 + 2y_P + \text{constant}$. Differentiating this equation twice with respect to time yields $0 = a_1 + 2a_P$. Since $a_P = a_2$, the relation is $a_1 + 2a_2 = 0$.