A boy weighing 50 kg runs up a staircase of 40 steps, each of 16 height cm, in 10 s. Calculate his power. (Take g = 10 m/s 2)
320 W
Calculating Power When Climbing Stairs
The problem asks us to calculate the power exerted by a boy while running up a staircase. Power is defined as the rate at which work is done. In this case, the work done by the boy is against gravity to increase his potential energy as he moves upwards.
Understanding the Physics Concepts
- Work Done: When an object is lifted against gravity, the work done is equal to the gain in gravitational potential energy. This is given by the formula: \(W = mgh\), where \(m\) is the mass, \(g\) is the acceleration due to gravity, and \(h\) is the vertical height lifted.
- Power: Power is the work done per unit of time. The formula for power is: \(P = \frac{W}{t}\), where \(W\) is the work done and \(t\) is the time taken.
Given Information
Let's list the information provided in the question:
- Mass of the boy, \(m = 50 \text{ kg}\)
- Number of steps = 40
- Height of each step, \(h_{\text{step}} = 16 \text{ cm}\)
- Time taken, \(t = 10 \text{ s}\)
- Acceleration due to gravity, \(g = 10 \text{ m/s}^2\)
Step-by-Step Calculation of Power
First, we need to find the total vertical height the boy climbs. The height of each step is given in centimeters, so we must convert it to meters.
Height of each step in meters: \(h_{\text{step}} = 16 \text{ cm} = \frac{16}{100} \text{ m} = 0.16 \text{ m}\)
Total height climbed, \(h_{\text{total}}\), is the number of steps multiplied by the height of each step:
\(h_{\text{total}} = \text{Number of steps} \times h_{\text{step}}\)
\(h_{\text{total}} = 40 \times 0.16 \text{ m}\)
\(h_{\text{total}} = 6.4 \text{ m}\)
Next, we calculate the work done by the boy against gravity to reach this height. Using the formula \(W = mgh\):
\(W = 50 \text{ kg} \times 10 \text{ m/s}^2 \times 6.4 \text{ m}\)
\(W = 500 \times 6.4 \text{ J}\)
\(W = 3200 \text{ J}\)
Finally, we can calculate the power using the formula \(P = \frac{W}{t}\):
\(P = \frac{3200 \text{ J}}{10 \text{ s}}\)
\(P = 320 \text{ W}\)
The power exerted by the boy is 320 Watts.
Summary of Calculation
| Quantity | Value | Unit | Calculation |
|---|---|---|---|
| Mass (m) | 50 | kg | Given |
| Steps | 40 | - | Given |
| Step Height (cm) | 16 | cm | Given |
| Step Height (m) | 0.16 | m | \(16 \text{ cm} \div 100\) |
| Total Height (h) | 6.4 | m | \(40 \times 0.16\) |
| Gravity (g) | 10 | m/s<sup>2</sup> | Given |
| Time (t) | 10 | s | Given |
| Work (W) | 3200 | J | \(m \times g \times h = 50 \times 10 \times 6.4\) |
| Power (P) | 320 | W | \(W \div t = 3200 \div 10\) |
Conclusion
Based on our calculations, the power exerted by the boy running up the staircase is 320 W.
Revision Table: Power, Work, and Energy
| Term | Definition | Formula | Units |
|---|---|---|---|
| Work (W) | Energy transferred when a force moves an object over a distance; done against gravity is potential energy change. | \(W = Fd\) (general), \(W = mgh\) (against gravity) | Joules (J) |
| Potential Energy (PE) | Energy stored by an object due to its position or state. Gravitational PE is due to height. | \(PE = mgh\) | Joules (J) |
| Power (P) | The rate at which work is done or energy is transferred. | \(P = \frac{W}{t}\) or \(P = \frac{\Delta E}{t}\) | Watts (W) |
Additional Information on Power Calculation
Power is a scalar quantity. It tells us how quickly energy is being used or transferred. A higher power means that work is being done faster. The unit of power, the Watt (W), is defined as 1 Joule per second (1 W = 1 J/s).
In this problem, the work done is against the force of gravity. The force due to gravity on the boy is his weight, which is \(F_g = mg\). When he climbs vertically, the work done against this force is \(W = F_g \times h = (mg)h\), assuming the force he exerts matches the gravitational force (plus a small amount to accelerate upwards, but the problem implies constant speed effectively by giving a total time). The power is then this work divided by the time taken.
Understanding the relationship between work, energy, and power is fundamental in physics. Work done results in a change in energy (like potential energy here), and power quantifies the rate of this energy change or work done.
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