A block A of mass m is connected by a light inextensible string to a second block B of mass 3 kg - Leaving Cert Applied Maths - Question 4 - 2018

Starting from the tension expression we can set up the following equations:

1. Substitute T = 3.92 N into the equation derived earlier: $mg \left( \frac{3}{5} \right) + T - \frac{2}{3} m g \left( \frac{4}{5} \right) = ma$

2. Rearranging gives us: $3.92 = mg \left( \frac{3}{5} \right) + T - \frac{2}{3} m g \left( \frac{4}{5} \right)$

3. Combining similar terms yields: $3.92 = \frac{mg(a + b)}{3 + m}$

4. Finally, solving for m leads to: $m = \frac{3.92 (3 + m)}{g}$. After carefully calculating the numerical values, we can find:

Thus the mass m is discovered to be 2 kg.

In the first diagram for the moveable pulley:

• The forces are the tensions T acting upwards on both sides, and the combined weight of mass m acting downwards:

                T
            +-----+
            |     m |
            +-----+
                |  
            mg

For the 6 kg mass:

• The tension T acts upwards while the weight of the mass acts downwards:

            T
            +-----+
            |  6kg |
            +-----+
                |  
            6g

For the 3 kg mass:

• Similarly, T acts upwards with the weight acting downwards:

            T
            +-----+
            |  3kg |
            +-----+
                |  
            3g

Using the free body diagrams established earlier, we can set up the following equations:

1. For the 6 kg mass: $T - 6g = 6a$

2. For the 3 kg mass: $T - 3g = 3b$

3. Combine and solve for tensions: $mg - 2T = \frac{1}{2} (a + b)$

4. Solving for T will give: $T = \frac{(6a + 3b)}{6}$

Finally, substitute the values to find T in terms of m.

To find the condition for the acceleration of the moveable pulley to be zero, we set: $\frac{1}{2}(a + b) = 0$

Given the motion equations lead to:

1. Setting mass m effectively cancels forces:
2. Solving for m leads to: $m = 8$. Therefore, the required mass for balance and no acceleration is m = 8 kg.