9. (a) A solid sphere floats at rest in water - Leaving Cert Applied Maths - Question 9 - 2015

To calculate the tension in the string, we first need to find the buoyant force acting on the cylinder. The buoyant force ($B$) can be calculated using:

$B = \text{Density}_{liquid} \times V_{cylinder} \times g$

Where:

• The volume of the cylinder can be computed as:

$V_{cylinder} = \pi r^2 h$

Given that:

• Radius $r = 0.04 ext{ m}$
• Height $h = 0.15 ext{ m}$
• Relative density of the liquid is $1.2$, thus:
• Density of the liquid is $\text{Density}_{liquid} = 1.2 \times 1000 = 1200 ext{ kg m}^{-3}$.

Calculating the volume:

$V_{cylinder} = \pi (0.04)^2 (0.15) \approx 0.0075 \text{ m}^3$.

Calculating the buoyant force:

$B = 1200 \times 0.0075 \times 10 \approx 90 \text{ N}$.

The weight of the cylinder ($W$) is:

• Relative density of the cylinder = 0.8 means its density is $800 ext{ kg m}^{-3}$, thus:

$W = 800 \times 0.0075 \times 10 = 60 \text{ N}$.

Using the equilibrium of forces for the cylinder:

$T + W = B$

Where:

• $T$ is the tension in the string. Solving for $T$:

$T = B - W = 90 - 60 = 30 ext{ N}$.

Thus, the tension in the string is 30 N.