Question

A force separately produces accelerations of 18 ms⁻², 9 ms⁻² and 6 ms⁻² in three bodies of masses P, Q and R respectively. If the same force is applied on a body of mass P+Q+R, then the acceleration of that body is

• A. 3 ms⁻²
• B. 6 ms⁻²
• C. 2 ms⁻²
• D. 33 ms⁻²

Hint:

For problems involving the same force acting on different masses, remember the inverse relationship between mass and acceleration (\(a \propto 1/m\)). The combined body will have a larger mass, so its acceleration must be smaller than the smallest individual acceleration (which was 6 ms⁻²). This can help eliminate some options quickly.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem is based on Newton's Second Law of Motion, which states that Force = mass \(\times\) acceleration (\(F=ma\)). We will use this relationship to find the individual masses in terms of the applied force, and then find the acceleration of the combined mass.
Step 2: Key Formula or Approach:
From \(F=ma\), we can express mass as \(m = F/a\). Let the constant force be \(F\).
Step 3: Detailed Explanation:
Let's find the masses P, Q, and R in terms of the force F.
For mass P: \(F = P \times a_P \implies F = P \times 18 \implies P = \frac{F}{18}\).
For mass Q: \(F = Q \times a_Q \implies F = Q \times 9 \implies Q = \frac{F}{9}\).
For mass R: \(F = R \times a_R \implies F = R \times 6 \implies R = \frac{F}{6}\).
Now, we find the total mass of the combined body, \(M_{total}\).
\[ M_{total} = P + Q + R = \frac{F}{18} + \frac{F}{9} + \frac{F}{6} \] To add these fractions, we find a common denominator, which is 18.
\[ M_{total} = F \left( \frac{1}{18} + \frac{2}{18} + \frac{3}{18} \right) = F \left( \frac{1+2+3}{18} \right) = F \left( \frac{6}{18} \right) = \frac{F}{3} \] The total mass of the system is \(\frac{F}{3}\).
Now, the same force \(F\) is applied to this total mass. Let the new acceleration be \(a_{new}\).
\[ F = M_{total} \times a_{new} \] \[ F = \left( \frac{F}{3} \right) \times a_{new} \] We can cancel \(F\) from both sides (assuming F is not zero).
\[ 1 = \frac{a_{new}}{3} \] \[ a_{new} = 3 \text{ ms⁻²} \] Step 4: Final Answer:
The acceleration of the combined body is 3 ms⁻². Thus, option (A) is correct.