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Step-by-Step Solution
Step 1: Identify the Relevant Physical Principles
The motion of a pendulum can be analyzed using the principle of conservation of mechanical energy, which states that the sum of kinetic energy (KE) and potential energy (PE) remains constant in the absence of non-conservative forces (like air resistance).
Step 2: Define the Reference for Potential Energy
Let the lowest position of the pendulum be the reference point where the potential energy is zero. At this position, the bob has maximum speed and no potential energy (PE = 0).
Step 3: Determine the Initial Energy at the Lowest Position
At the lowest position:
- Speed = 3 \text{ m/s}
- Kinetic energy = \frac{1}{2} m v^2 = \frac{1}{2} m (3)^2 = \frac{9}{2} m = 4.5 \, m joules (where m is the mass of the bob).
- Potential energy = 0 (as chosen reference).
Therefore, total mechanical energy at the lowest position ( E_\text{initial} ):
E_\text{initial} = 4.5\, m + 0 = 4.5 \, m.
Step 4: Calculate the Height Gain When the Bob Reaches 60°
When the pendulum swings to an angle of 60° from the vertical, it rises to a height h relative to the lowest point. This height is given by:
h = l - l\cos(60^\circ) = l \bigl[1 - \cos(60^\circ)\bigr].
Given l = 0.50 \text{ m} and \cos(60^\circ) = \frac{1}{2} , we have:
h = 0.50 - 0.50 \times \frac{1}{2} = 0.50 - 0.25 = 0.25 \text{ m}.
The corresponding potential energy increase is:
PE = mgh = m \times 10 \times 0.25 = 2.5 \, m \text{ joules}.
Step 5: Apply Conservation of Mechanical Energy
As the pendulum swings from the bottom to 60°, the total energy remains the same:
E_\text{initial} = E_\text{final}.
E_\text{final} = KE_{\text{at }60^\circ} + PE_{\text{at }60^\circ}.
So, 4.5\, m = \frac{1}{2} m v^2 + 2.5\, m.
Dividing through by m (assuming m \neq 0 ):
4.5 = \frac{1}{2} v^2 + 2.5.
\frac{1}{2} v^2 = 4.5 - 2.5 = 2.
v^2 = 4.
v = 2 \text{ m/s}.
Step 6: State the Final Answer
The speed of the pendulum bob when it makes an angle of 60° with the vertical is 2 m/s.
