if the body describes vertical circle then the minimum at an angle theta from the lowest position is
To find the minimum height at an angle theta from the lowest position in a vertical circle, we need to consider the conservation of mechanical energy.
The mechanical energy of the system is the sum of the kinetic energy (KE) and potential energy (PE). At the lowest position, when the body is at the bottom of the vertical circle, all of the system's energy is in the form of kinetic energy. As the body moves up the circle, some of the kinetic energy is converted to potential energy, and vice versa.
At any point in the vertical circle, the mechanical energy remains constant. Therefore, we can equate the initial mechanical energy at the lowest position to the sum of kinetic and potential energy at the point of interest.
Setting up this equation, we have:
KE_initial = KE + PE
At the lowest position, all the energy is in the form of kinetic energy. So KE_initial = KE.
At an angle theta from the lowest position, the body has moved upwards, and some of the kinetic energy has been converted to potential energy. The potential energy at this point is given by PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height above the lowest point.
Substituting these values into the equation, we get:
KE_initial = KE + mgh
Simplifying the equation using the forms of kinetic and potential energy for a rotating body (KE = (1/2)mv^2 and h = R - Rcos(theta), where v is the velocity and R is the radius of the circle), we have:
(1/2)mV_initial^2 = (1/2)mv^2 + mg(R - Rcos(theta))
Here, V_initial is the initial velocity at the lowest position and v is the velocity at the point of interest.
Rearranging the equation, we can solve for v:
V_initial^2 = v^2 + 2gR(1 - cos(theta))
Now, we can find the minimum height at an angle theta by determining the value of h using the equation for potential energy:
PE = mgh
Since the point of interest is at the minimum height, the potential energy is at its minimum value, which occurs when h is minimized. Thus, we need to find the smallest value of h.
Using the expression for h in terms of theta (h = R - Rcos(theta)), we can substitute it into the equation:
PE = mg(R - Rcos(theta))
Now, the minimum potential energy occurs when cos(theta) is maximized (which happens when theta = 0, cos(0) = 1). So, substituting theta = 0 into the equation, we get:
PE_min = mg(R - Rcos(0))
PE_min = mg(R - R)
PE_min = mg(0)
PE_min = 0
Therefore, the minimum height at angle theta from the lowest position is 0, which means the body is at its lowest point in the vertical circle.