A 50 kg girl is swinging on a swing from rest. Then, the power delivered when moving with a velocity of `2ms^(-1)` upwards in a direction making an angle `60^(@)` with the vertical is

Options

A. 490`sqrt3W`
B. 490 W
C. `490sqrt2W`
D. 245 W

Explanation

To solve the problem, we need to calculate the power delivered when the girl is swinging on the swing. The power can be calculated using the formula:

\[ P = F \cdot v \cdot \cos(\theta) \]

Where:
- \( P \) is the power,
- \( F \) is the force contributing to the power,
- \( v \) is the velocity,
- \( \theta \) is the angle between the force and the direction of motion.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the girl, \( m = 50 \, \text{kg} \)
- Velocity, \( v = 2 \, \text{m/s} \)
- Angle with the vertical, \( \theta = 60^\circ \)

2. **Calculate the Weight of the Girl:**
The weight \( W \) can be calculated using the formula:
\[ W = m \cdot g \]
Where \( g \) (acceleration due to gravity) is approximately \( 9.8 \, \text{m/s}^2 \).
\[ W = 50 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 490 \, \text{N} \]

3. **Determine the Angle between the Weight and the Velocity:**
The angle between the weight \( W \) (acting vertically downwards) and the velocity \( v \) (which is at an angle of \( 60^\circ \) with the vertical) is:
\[ \text{Angle} = 90^\circ + 60^\circ = 150^\circ \]

4. **Calculate the Power Delivered:**
Using the power formula:
\[ P = W \cdot v \cdot \cos(150^\circ) \]
We know that:
\[ \cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2} \]
Therefore, substituting the values:
\[ P = 490 \, \text{N} \cdot 2 \, \text{m/s} \cdot \left(-\frac{\sqrt{3}}{2}\right) \]
Simplifying this:
\[ P = 490 \cdot 2 \cdot \left(-\frac{\sqrt{3}}{2}\right) = 490 \cdot (-\sqrt{3}) = -490\sqrt{3} \, \text{W} \]

5. **Final Result:**
The power delivered when the girl is swinging is:
\[ P = 490\sqrt{3} \, \text{W} \quad (\text{taking the magnitude}) \]

### Conclusion:
The power delivered when moving with a velocity of \( 2 \, \text{m/s} \) upwards at an angle of \( 60^\circ \) with the vertical is approximately \( 490\sqrt{3} \, \text{W} \).

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