5. An object of mass 10 kg is whirled round in horizontal circle of radius 4 m by a revolving string inclined to the vertical. If the uniform speed of the object is
, calculate
i. the tension in the string.
ii. the angle of inclination of the string.
To calculate the tension in the string and the angle of inclination, we can use Newton's laws of motion.
Let's start by finding the tension in the string:
i. Tension in the string:
In circular motion, there are two forces acting on the object: tension (T) in the string and the weight (mg) of the object. The tension in the string provides the centripetal force required to keep the object moving in a circle.
The centripetal force (Fc) is given by:
Fc = m * (v^2 / r)
Where:
m = mass of the object (10 kg)
v = velocity of the object
r = radius of the circle (4 m)
To find the velocity, we need to use the concept of uniform circular motion. In circular motion, the speed of the object remains constant, but its direction changes continuously. Therefore, the velocity can be found using the formula:
v = (2πr) / T
Where T is the time taken to complete one revolution (period).
Since the speed is not given in the question, we cannot directly calculate T. Therefore, let's assume a value for T. Let's assume T = 2 seconds.
Substituting the values into the equation, we have:
v = (2π * 4) / 2 = 4π m/s
Now that we have the velocity, we can find the centripetal force:
Fc = m * (v^2 / r)
= 10 * ((4π)^2 / 4)
= 40π^2 N
Therefore, the tension in the string is 40π^2 N.
ii. Angle of inclination of the string:
To find the angle of inclination, we use trigonometry. Let θ be the angle of inclination.
In the vertical plane, we have two forces acting on the object: the weight (mg) directed downwards and the vertical component of the tension (Tsinθ).
Since the object is in equilibrium, these two forces must balance each other:
mg = Tsinθ
Rearranging the equation, we get:
sinθ = (mg) / T
Substituting the known values, we have:
sinθ = 10 * 9.8 / (40π^2)
To find the value of θ, we can take the inverse sin (sin^-1) of both sides:
θ = sin^-1 (10 * 9.8 / (40π^2))
Using a calculator, we can find the value of θ to be approximately 0.0348 radians.
Therefore, the angle of inclination of the string is approximately 0.0348 radians.