A mass of 8kg is whirled round in a vertical circle using a rope of length 80cm if it makes 2.5 cycles in 1 second, calculate the maximum tension the rope experiences

Answer

Yes

To calculate the maximum tension the rope experiences while whirling the mass in a vertical circle, we need to consider two main forces acting on the mass: the weight (mg) and the tension in the rope (T).

In a vertical circular motion, the tension in the rope will vary throughout the motion. The maximum tension occurs at the bottom of the circle when the mass is at its lowest point.

To find the maximum tension, we'll use the centripetal force equation:

F = mv^2 / r

Where:
F is the centripetal force (which is equal to the tension in the rope at the bottom of the circle),
m is the mass,
v is the velocity, and
r is the radius of the circle.

First, let's find the velocity of the mass. Since 2.5 cycles are made in 1 second, we can calculate the time taken for one complete cycle (T) by dividing the total time taken (1 second) by the number of cycles (2.5):

T = 1 second / 2.5 cycles = 0.4 seconds

The velocity (v) can be found using the formula:

v = 2πr / T

Given that the radius (r) is 80 cm, we need to convert it to meters by dividing by 100:

r = 80 cm / 100 = 0.8 meters

Now we can substitute the values into the equation to find the velocity:

v = 2π(0.8) / 0.4 = 4π m/s

Next, we can calculate the maximum tension (T) by using the centripetal force equation:

T = m(4π)^2r / r

Let's substitute the values:

T = 8kg * (4π)^2 * 0.8m / 0.8m

Simplifying the equation, we can cancel out the common factors:

T = 8kg * (16π^2) = 128π^2 kg

Now, to find the numerical value of the maximum tension, we can use an approximation for π:

T ≈ 128 * 3.14^2 kg

Evaluating the expression, we find:

T ≈ 1258.28 kg

Therefore, the maximum tension the rope experiences is approximately 1258.28 kg.