a small body of mass m revolving in a horizontal circle with velocity V of a constant magnitude at the end of a cord of length L. As the body swings around its path, the cord sweeps over the surface of a cone. The cord makes an angle è with the vertical, so the radius of the circle in which the body moves is R=L sin è and the magnitude of the velocity V, equals V= 2ᴫr/T = 2ᴫLsin è/T where T is the period of the revolution of the motion, the time for one complete revolution. Find T
To find the period T, we can start by using the equation for the magnitude of velocity V:
V = (2πR) / T
Substituting R = Lsinθ, we have:
V = (2πLsinθ) / T
Now, let's isolate T:
T = (2πLsinθ) / V
Therefore, the period of revolution T is given by the equation T = (2πLsinθ) / V.
To find the period T of the revolution motion given the velocity V and radius R, we can use the formula:
V = 2πR / T
In this case, we are given that the velocity V equals 2πLsin(θ) / T, where L is the length of the cord and θ is the angle the cord makes with the vertical.
By comparing the two equations, we can equate them and solve for T:
2πLsin(θ) / T = 2πR / T
Since R = Lsin(θ), we can substitute that in the equation:
2πLsin(θ) / T = 2π(Lsin(θ)) / T
The Lsin(θ) terms cancel out:
2π / T = 2π / T
This equation holds true for any value of T. Thus, T can be any positive real number.
Therefore, the period T of the revolution motion is not uniquely determined by the given information. It can have any positive value.