A mass M is attached to the end of a rope of length 90cm. If the tension in the rope is 4000N when the mass is moving at 360 rev/min, calculate the value of M.

Steps to Solve

1. Convert revolutions per minute (RPM) to radians per second (rad/s)

To do this, we use the conversion factor $1 , \text{rev} = 2\pi , \text{radians}$ and $1 , \text{minute} = 60 , \text{seconds}$.

Given the speed is 30 RPM, we convert as follows:

$$ \omega = 30 \frac{\text{rev}}{\text{min}} \times \frac{2\pi , \text{rad}}{1 , \text{rev}} \times \frac{1 , \text{min}}{60 , \text{s}} = \pi , \text{rad/s} $$

2. Determine the centripetal force equation

The centripetal force ($F_c$) is given by the equation:

$$ F_c = M \cdot a_c = M \cdot \frac{v^2}{r} $$

Where $M$ is the mass, $a_c$ is the centripetal acceleration, $v$ is the velocity, and $r$ is the radius of the circular path. We also know that $v = r\omega$, therefore:

$$ F_c = M \cdot r \cdot \omega^2 $$

3. Solve for M

We are given the centripetal force as the tension in the rope, $F_c = 40 , \text{N}$, and the radius $r = 2 , \text{m}$. We also calculated $\omega = \pi , \text{rad/s}$. Now, we can rearrange the formula to solve for the mass $M$:

$$ M = \frac{F_c}{r \cdot \omega^2} $$

4. Calculate the mass M

Substitute the given values into the equation:

$$ M = \frac{40 , \text{N}}{2 , \text{m} \cdot (\pi , \text{rad/s})^2} $$

$$ M = \frac{40}{2\pi^2} , \text{kg} $$

$$ M \approx 2.026 , \text{kg} $$