3 four pulleys equally spaced along a shaft balance a mass. What is the radius of the balance mass at the second pulley?

Steps to Solve

1. Identify the masses and forces involved

Determine the masses of the objects involved in the pulley system. Assign variables to the masses for clarity; for example, let $m_1$ be the mass on one side, and $m_2$ be the mass on the other side.

2. Apply Newton's Second Law

According to Newton's Second Law, the net force ($F_{net}$) acting on an object is equal to the mass multiplied by the acceleration ($a$). For each mass, set up the equations as follows:

• For mass $m_1$: $$ F_{net, m_1} = m_1 g - T = m_1 a $$
• For mass $m_2$: $$ F_{net, m_2} = T - m_2 g = m_2 (-a) $$

Here, $g$ is the acceleration due to gravity, and $T$ is the tension in the string.

3. Solve the system of equations

Combine the two equations derived from Newton's Second Law. You can eliminate tension ($T$) by solving for it in one equation and substituting it into the other, resulting in a single equation in terms of $a$. Solve for $a$:

$$ m_1 g - m_2 g = (m_1 + m_2) a $$

4. Rearranging for acceleration

Isolate $a$ to find the acceleration of the system:

$$ a = \frac{(m_1 - m_2)g}{m_1 + m_2} $$

This gives the acceleration of both masses.

5. Determine the tension in the string

Once you have the acceleration, substitute $a$ back into one of the original equations to find the tension ($T$) in the string. Use the equation for either mass:

$$ T = m_1 g - m_1 a $$

6. Check the results

Finally, verify your calculations by substituting the values back into the context of the problem and ensuring the forces and tension balance out according to the laws of physics.