What is the speed of the bob at the lowest point of its swing? (Use g = 9.8 m/s²)
Understand the Problem
The question is asking for the calculation of the speed of a pendulum bob at the lowest point of its swing, given its mass and the height from which it was released. This involves applying principles of energy conservation, particularly converting potential energy into kinetic energy.
Answer
The speed of the bob at the lowest point of its swing is $7 \, \text{m/s}$.
Answer for screen readers
The speed of the bob at the lowest point of its swing is $7 , \text{m/s}$.
Steps to Solve
- Identify Given Values
The mass of the pendulum bob, $m = 2 , \text{kg}$, and the height from which it was released, $h = 2.5 , \text{m}$. The acceleration due to gravity is given as $g = 9.8 , \text{m/s}^2$.
- Calculate Potential Energy at the Starting Height
The potential energy (PE) at the height can be calculated using the formula: $$ \text{PE} = m \cdot g \cdot h $$
Substituting the given values: $$ \text{PE} = 2 , \text{kg} \cdot 9.8 , \text{m/s}^2 \cdot 2.5 , \text{m} $$
- Calculate the Potential Energy
Now, let's compute the potential energy: $$ \text{PE} = 2 \cdot 9.8 \cdot 2.5 = 49 , \text{J} $$
- Apply Conservation of Energy Principle
At the lowest point, all potential energy converts to kinetic energy (KE). Thus, we have: $$ \text{KE} = \text{PE} $$
- Set Kinetic Energy Formula
The kinetic energy is defined as: $$ \text{KE} = \frac{1}{2} m v^2 $$
Where (v) is the speed we want to find.
- Equate Kinetic Energy and Potential Energy
From the conservation of energy: $$ \frac{1}{2} m v^2 = \text{PE} $$
Substituting the potential energy we calculated: $$ \frac{1}{2} m v^2 = 49 , \text{J} $$
- Solve for the Speed (v)
Rearranging the equation to solve for (v): $$ v^2 = \frac{2 \cdot 49}{m} $$
Substituting (m = 2 , \text{kg}): $$ v^2 = \frac{2 \cdot 49}{2} = 49 $$
Then taking the square root: $$ v = \sqrt{49} = 7 , \text{m/s} $$
The speed of the bob at the lowest point of its swing is $7 , \text{m/s}$.
More Information
The conservation of energy principle states that the total mechanical energy remains constant if only conservative forces, like gravity, are acting. This is why the potential energy converts entirely to kinetic energy at the lowest point of the pendulum swing.
Tips
- Forgetting to convert units: Always ensure that height is in meters, mass in kilograms, and acceleration due to gravity in the appropriate units.
- Neglecting energy losses: In practical scenarios like air resistance, energy may not convert perfectly, but here we assume ideal conditions.
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