Three blocks of masses \( m_1 = 2\text{ kg} \), \( m_2 = 3\text{ kg} \) and \( m_3 = 5\text{ kg} \) are placed on a horizontal frictionless surface and a force of \( 30\text{ N} \) pulls the system as shown below. The value of tension \( T \) will be
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In connected body systems, the tension in any string is always equal to the sum of all masses behind it multiplied by the common acceleration of the system. This allows you to find any tension instantly without drawing complete free-body diagrams.
Step 1: Understanding the Concept:
When multiple masses are connected by strings and pulled by an external force on a frictionless surface, the entire system moves with a common acceleration.
First, we find this common acceleration by treating the whole set of blocks as a single system.
Then, to find the tension in any specific segment of the string, we analyze the free-body diagram of the specific block(s) being pulled by that tension. Step 2: Key Formula or Approach:
Common acceleration: \[a = \frac{F_{total}}{M_{total}}\]
Newton's Second Law for a single block: \[F_{net} = m \cdot a\] Step 3: Detailed Explanation:
The total mass of the system is:
\[M_{total} = m_1 + m_2 + m_3 = 2 + 3 + 5 = 10 \text{ kg}\]
The external pulling force is \(F = 30 \text{ N}\).
The common acceleration \(a\) of all three blocks is:
\[a = \frac{F}{M_{total}} = \frac{30}{10} = 3 \text{ m/s}^2\]
The question asks for the tension \(T\) in the string. From the diagram, it is observed that the tension \(T\) is between \(m_1\) and \(m_2\).
Actually, looking at the layout, \(m_1\) is at the trailing end, connected to \(m_2\), which is connected to \(m_3\), which is pulled by 30 N.
The tension \(T\) between \(m_1\) and \(m_2\) is responsible for accelerating ONLY block \(m_1\).
Applying Newton's second law for \(m_1\):
\[T = m_1 \times a\]
\[T = 2 \text{ kg} \times 3 \text{ m/s}^2 = 6 \text{ N}\]
Note: If the tension between \(m_2\) and \(m_3\) was asked, it would be \((m_1 + m_2) \cdot a = (2+3) \cdot 3 = 15 \text{ N}\).
But for the tension pulling the last block \(m_1\), it is 6 N. Step 4: Final Answer:
The common acceleration is 3 m/s\(^2\). The tension \(T\) required to pull the 2 kg block at this acceleration is 6 N.
