A 3 kg mass attached to a string 6 m long is to be swung in a circular at a constant speed making one complete revolution in 1.25 s. Determine the value that the breaking strain that the string must not exceed if the string is not to break when the circular motion is in

(a) a horizontal plane?
(b) avertical plane?

To determine the value of the breaking strain that the string must not exceed in both the horizontal and vertical planes, we need to consider the forces acting on the mass in circular motion.

In circular motion, the centripetal force is responsible for keeping an object moving in a circular path. It is given by the equation:

Fc = (m * v^2) / r

where Fc is the centripetal force, m is the mass, v is the velocity, and r is the radius of the circular path.

(a) Horizontal Plane:
In the horizontal plane, the tension in the string provides the necessary centripetal force to keep the mass moving in a circle. So, we can equate the tension in the string to the centripetal force:

T = (m * v^2) / r

Given that the mass (m) is 3 kg, the radius (r) is 6 m, and the time taken for one complete revolution is 1.25 s, we need to find the velocity (v) first.

To get the velocity, we can use the formula for speed:

Speed = Distance / Time

In one complete revolution, the distance is equal to the circumference of the circle, which is equal to 2πr.

Speed = (2πr) / Time

Substituting the values, we get:

Speed = (2π * 6) / 1.25

Next, we can substitute the calculated value of speed into the equation for tension:

T = (3 * ([(2π * 6) / 1.25])^2) / 6

Simplifying the equation gives us the tension in the horizontal plane.

(b) Vertical Plane:
In the vertical plane, we need to consider the weight of the mass in addition to the centripetal force. The tension in the string must be able to support the weight of the mass and provide the centripetal force.

The weight (W) is given by the equation:

W = m * g

where g is the acceleration due to gravity (approximately 9.81 m/s^2).

To find the tension in the vertical plane, we can equate the sum of the weight and the centripetal force to the tension:

T + W = (m * v^2) / r

Substituting the values, we have:

T + (3 * 9.81) = (3 * ([(2π * 6) / 1.25])^2) / 6

After calculating the value of tension, we subtract the weight from it to find the actual breaking strain.

It's important to note that the breaking strain of the string depends on the material it is made of, and this value should be provided by the manufacturer or determined through testing.