Question

Three masses $m_1 = 4$ kg, $m_2 = 4$ kg and $m_3 = 6$ kg are suspended from a fixed smooth frictionless pulley as shown in the figure below. The value of $T_1/T_2$ is: (take $g = 10 \text{ m/s}^2$)}

• A. 5/3
• B. 2/3
• C. 3/5
• D. 3/5

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We have an Atwood machine with a compound mass on one side. We need to find the tensions in the main string ($T_1$) and the string connecting the two masses on the right side ($T_2$), and then find their ratio.
Step 2: Key Formula or Approach:
1. System Acceleration: First, treat the entire system as one and find the common acceleration using $a = \frac{\text{Net Driving Force}}{\text{Total Mass}}$.
2. Tension Calculation: Isolate individual masses (or groups of masses) and apply Newton's Second Law ($F_{net} = ma$) to find the tensions.
Step 3: Detailed Explanation:
Part 1: Find the acceleration of the system
The mass on the left side is $m_1 = 4$ kg.
The total mass on the right side is $M_R = m_2 + m_3 = 4 + 6 = 10$ kg.
Since $M_R>m_1$, the right side will accelerate downwards and the left side will accelerate upwards.
The net driving force is the difference in weights: $F_{net} = M_R g - m_1 g = (10 - 4)g = 6g$.
The total mass being accelerated is $M_{total} = m_1 + m_2 + m_3 = 4 + 4 + 6 = 14$ kg.
The acceleration of the system is:
\[ a = \frac{F_{net}}{M_{total}} = \frac{6g}{14} = \frac{3g}{7} \] Part 2: Calculate the tensions
To find $T_1$, we can analyze the motion of mass $m_1$. It accelerates upwards.
Applying Newton's Second Law to $m_1$:
$T_1 - m_1 g = m_1 a \implies T_1 = m_1(g+a)$.
\[ T_1 = 4 \left( g + \frac{3g}{7} \right) = 4 \left( \frac{10g}{7} \right) = \frac{40g}{7} \] To find $T_2$, we analyze the motion of mass $m_3$. It accelerates downwards.
Applying Newton's Second Law to $m_3$:
$m_3 g - T_2 = m_3 a \implies T_2 = m_3(g-a)$.
\[ T_2 = 6 \left( g - \frac{3g}{7} \right) = 6 \left( \frac{4g}{7} \right) = \frac{24g}{7} \] Part 3: Find the ratio
\[ \frac{T_1}{T_2} = \frac{40g/7}{24g/7} = \frac{40}{24} \] Simplifying the fraction by dividing by 8:
\[ \frac{T_1}{T_2} = \frac{5}{3} \] Step 4: Final Answer:
The ratio $T_1/T_2$ is 5/3.