(a) A particle describes a horizontal circle of radius 2 m with uniform angular velocity ω radians per second - Leaving Cert Applied Maths - Question 8 - 2011

The time taken to complete one revolution can be found using the following formula:

$T = \frac{2\pi r}{v}$

Where:

• $r$ is the radius (2 m)
• $v$ is the speed of the particle (8 m/s)

Calculating the time:

$T = \frac{2\pi (2)}{8} = \frac{4\pi}{8} = \frac{\pi}{2} \text{ seconds}$

Thus, the time taken to complete one revolution is ( \frac{\pi}{2} ) seconds.

To find the tension in the string, we will analyze the forces acting on the particle in circular motion under the angle α. Using the formula:

$T \sin(\alpha) = \frac{mv^2}{r}$

Where:

• $m$ is the mass (3 kg)
• $v$ is the speed (2 m/s)
• $r$ is the radius (0.5 m)
• $\tan(\alpha) = \frac{4}{3} \Rightarrow \sin(\alpha) = \frac{4}{5}$ and $\cos(\alpha) = \frac{3}{5}$

Substituting into the equation gives:

$T \frac{4}{5} = \frac{3(2)^2}{0.5} \Rightarrow T \frac{4}{5} = \frac{12}{0.5} = 24$

Thus, we can solve for $T$:

$T = 24 \cdot \frac{5}{4} = 30 \, \text{N}$

Therefore, the tension in the string is 30 N.

To find the reaction force, we use the vertical forces acting on the particle. The equations can be written as:

R + 30 \cdot \frac{3}{5} = 3g \\ R + 18 = 9.81 \\ R = 9.81 - 18 \\ R = 12 \, \text{N} $$ Thus, the reaction force between the particle and the table is 12 N.