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When a force F is applied to a mass m(1)...

When a force F is applied to a mass `m_(1)`, its acceleration is `6" m"//"s"^(2)`. If the same force is applied to another mass `m_(2)`, it gives an acceleration `3" m"//"s"^(2)`. If the two masses are tied together and if the same force is applied to the combination it gives an acceleration

A

`1" m"//"s"^(2)`

B

`3" m"//"s"^(2)`

C

`2" m"//"s"^(2)`

D

`1.5" m"//"s"^(2)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will analyze the forces and accelerations involved for the two masses, and then determine the acceleration when they are tied together. ### Step 1: Understand the relationship between force, mass, and acceleration According to Newton's second law of motion, the relationship is given by: \[ F = m \cdot a \] Where: - \( F \) is the force applied, - \( m \) is the mass, - \( a \) is the acceleration. ### Step 2: Write down the equations for the two masses 1. For mass \( m_1 \): \[ F = m_1 \cdot 6 \] (since the acceleration is \( 6 \, \text{m/s}^2 \)) 2. For mass \( m_2 \): \[ F = m_2 \cdot 3 \] (since the acceleration is \( 3 \, \text{m/s}^2 \)) ### Step 3: Set the two equations equal to each other Since both equations represent the same force \( F \): \[ m_1 \cdot 6 = m_2 \cdot 3 \] ### Step 4: Solve for the relationship between \( m_1 \) and \( m_2 \) Rearranging the equation gives: \[ 6m_1 = 3m_2 \] Dividing both sides by 3: \[ 2m_1 = m_2 \] This shows that \( m_2 \) is twice the mass of \( m_1 \): \[ m_2 = 2m_1 \] ### Step 5: Calculate the total mass when both masses are tied together When \( m_1 \) and \( m_2 \) are tied together, the total mass \( M \) is: \[ M = m_1 + m_2 = m_1 + 2m_1 = 3m_1 \] ### Step 6: Write the equation for the combined mass under the same force Using the total mass \( M \) and the same force \( F \): \[ F = M \cdot A \] Substituting \( M \): \[ F = 3m_1 \cdot A \] ### Step 7: Substitute \( F \) from the first equation From the first equation for \( m_1 \): \[ F = 6m_1 \] Now we can set the two expressions for \( F \) equal to each other: \[ 6m_1 = 3m_1 \cdot A \] ### Step 8: Solve for acceleration \( A \) Dividing both sides by \( 3m_1 \): \[ A = \frac{6m_1}{3m_1} = 2 \, \text{m/s}^2 \] ### Final Answer The acceleration when the two masses are tied together and the same force is applied is: \[ A = 2 \, \text{m/s}^2 \] ---
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