Let's break down the situation you've described with the pendulum moving in a horizontal circle. You're right to consider both kinetic energy (KE) and potential energy (PE) when analyzing the total energy of the system. Let's clarify how these energies work together in this scenario.
Understanding Kinetic Energy in Circular Motion
You've correctly identified that the centripetal force acting on the pendulum is given by the equation:
mv²/r = -k/r²
This implies that:
mv² = -k/r
From this, we can express the kinetic energy (KE) of the pendulum as:
KE = 1/2 mv²
Substituting the expression for mv², we find:
KE = 1/2 (-k/r)
This leads to:
KE = -k/2r
Potential Energy Considerations
Now, regarding potential energy (PE), you mentioned that it is equal to -mgr when taking the lowest point of the pendulum's swing as the reference point. This is a crucial aspect of energy conservation in this system.
In a gravitational field, the potential energy of an object at height h is given by:
PE = mgh
In your case, if we consider the reference point at the lowest position of the pendulum, the height h can be expressed in terms of the string length r. If the pendulum is at an angle θ from the vertical, the height h can be calculated as:
h = r(1 - cos(θ))
Thus, the potential energy becomes:
PE = mg(r(1 - cos(θ)))
Total Energy of the System
The total mechanical energy (E) of the pendulum is the sum of its kinetic and potential energies:
E = KE + PE
Substituting the expressions we derived:
E = -k/2r + mg(r(1 - cos(θ)))
This total energy will remain constant if we assume no non-conservative forces (like air resistance) are acting on the system. The interplay between KE and PE will allow the pendulum to swing back and forth, converting energy from one form to another.
Final Thoughts
In summary, while you've correctly calculated the kinetic energy, it's essential to include the potential energy to fully understand the total energy of the pendulum system. The total energy will depend on both the kinetic energy derived from the centripetal force and the potential energy associated with the height of the pendulum in the gravitational field. This holistic view allows for a comprehensive understanding of the dynamics at play in your pendulum scenario.
