The following diagram shows a pendulum at different positions. Which one of the following statement is true?
The pendulum has minimum potential energy at position R.
Understanding Pendulum Potential Energy and Minimum Height
The question asks about the position where a pendulum has minimum potential energy. Potential energy, specifically gravitational potential energy in this case, is energy stored due to an object's position relative to a reference point, usually the ground or the lowest point in the system.
The formula for gravitational potential energy (\(PE\)) is:
\(PE = mgh\)
Where:
- \(m\) is the mass of the object (the pendulum bob)
- \(g\) is the acceleration due to gravity
- \(h\) is the height of the object above a chosen reference point
From the formula, we can see that potential energy is directly proportional to the height (\(h\)). This means the higher an object is, the greater its potential energy, and the lower it is, the smaller its potential energy.
Analyzing the Pendulum Diagram and Positions
Let's look at the diagram showing the pendulum at different positions:
| Position | Description relative to swing |
|---|---|
| P | Highest point on the left side of the swing |
| Q | Intermediate point between P and R |
| R | Lowest point of the swing (equilibrium position) |
| S | Intermediate point between R and T |
| T | Highest point on the right side of the swing |
The potential energy is minimum when the height (\(h\)) of the pendulum bob is minimum. In the diagram, the lowest point the pendulum bob reaches during its swing is position R.
Identifying the Position of Minimum Potential Energy
Comparing the heights of the pendulum bob at the different positions:
- Positions P and T are the highest points reached by the pendulum. Therefore, the potential energy is maximum at P and T.
- Positions Q and S are at intermediate heights between the highest points (P, T) and the lowest point (R). The potential energy at Q and S is less than at P and T but greater than at R.
- Position R is the lowest point in the pendulum's swing. At this point, the height (\(h\)) of the bob is minimum relative to any other point in its path. Consequently, the gravitational potential energy (\(mgh\)) is minimum at position R.
Conclusion on Pendulum's Potential Energy
Based on the relationship between potential energy and height, and observing the diagram, the pendulum has the minimum potential energy at the position where it is lowest. This corresponds to position R in the given diagram.
Revision Table: Pendulum Energy Summary
| Position | Height relative to lowest point | Potential Energy | Kinetic Energy | Total Mechanical Energy |
|---|---|---|---|---|
| P & T (Highest points) | Maximum | Maximum | Minimum (close to zero) | Constant (neglecting friction) |
| Q & S (Intermediate points) | Intermediate | Intermediate | Intermediate | Constant (neglecting friction) |
| R (Lowest point) | Minimum (zero if reference is at R) | Minimum | Maximum | Constant (neglecting friction) |
Additional Information: Energy Conservation in Pendulum Swing
In an ideal simple pendulum swing (neglecting air resistance and friction), the total mechanical energy (the sum of potential energy and kinetic energy) remains constant. As the pendulum swings, energy is continuously converted between potential energy and kinetic energy. When the pendulum is at its highest point (maximum height), its speed is momentarily zero (or minimum), so kinetic energy is minimum, and potential energy is maximum. As it swings downwards, its height decreases (potential energy decreases), and its speed increases (kinetic energy increases). At the lowest point, height is minimum (potential energy minimum), and speed is maximum (kinetic energy maximum).
This conversion demonstrates the principle of conservation of mechanical energy.
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