Two masses connected in series with two massless strings are hanging from a support as shown in the figure. Find the tension in the upper string.
Show Hint
When two masses are connected in series with massless strings, the tension in the upper string is the sum of the weights of both masses. This principle is applicable when the system is in equilibrium.
Step 1: Understanding the Concept:
In a vertically hanging system in equilibrium, each segment of the string must support the entire weight of all the masses located below it.
The net force acting on any part of the system is zero. Step 2: Key Formula or Approach:
The tension \(T\) in the string supporting a mass system is balanced by the gravitational force: \( T = \sum (m) \times g \).
We simply sum the masses below the specific string segment. Step 3: Detailed Explanation:
Let us break down the forces in the system.
The lower string only carries the weight of the lower mass, \(m_{2}\).
Therefore, the tension in the lower string is \( T_{\text{lower}} = m_{2}g \).
The upper string, however, connects the support to both mass \(m_{1}\) and mass \(m_{2}\).
It must support the combined weight of the entire system hanging below it.
Summing the forces vertically for the upper string gives:
\[ T_{\text{upper}} = (m_{1} + m_{2})g \]
Step 4: Final Answer:
The tension in the upper string is equal to \((m_{1} + m_{2})g\).
