maximum tension in the string of an oscillating pendulum is double of the minimum tension. Then angular amplitude is
To find the angular amplitude of an oscillating pendulum when the maximum tension in the string is double the minimum tension, we can use the principle of conservation of mechanical energy.
The mechanical energy of a pendulum is given by the sum of its potential energy and kinetic energy.
At the maximum point of the pendulum's oscillation, when the string is most stretched, the potential energy is maximum and the kinetic energy is minimum. Similarly, at the minimum point, the potential energy is minimum and the kinetic energy is maximum.
Since we know that the maximum tension in the string is double the minimum tension, it means that the potential energy at the maximum point is double the potential energy at the minimum point.
Now, the potential energy of a pendulum is proportional to the square of the angular displacement. Therefore, if the potential energy at the maximum point is double the potential energy at the minimum point, the square of the angular displacement at the maximum point should be double the square of the angular displacement at the minimum point.
Let's denote the angular amplitude by θ.
According to the proportionality mentioned above, we have:
θ^2 (at the maximum point) = 2 * θ^2 (at the minimum point)
Simplifying this equation, we get:
θ^2 = 2 * θ^2 / 2
θ^2 = θ^2
This equation is true for all non-zero values of θ.
Therefore, we can conclude that the angular amplitude of the pendulum can be any non-zero value.
Tension at the bottom is
mg+mv^2/r
Tension at the high point is
mg*cosTheta
2*mgCosTheta=mg+mv^2/r
but 1/2 mv^2= mgr(1-cosTheta)
2mgCosTheta=mg+2mg(1-cosTheta)
cosTheta(2mg+2mg)=mg
cosTheta= 1/4
Theta= arcCos .25
And that is amplitude.