Question

How much work is done when a 75.0 -kg person climbs a flight of stairs \(10.0 \mathrm{~m}\) high at constant speed? a) \(7.36 \cdot 10^{5} \mathrm{~J}\) b) 750 J c) 75 J d) \(7500 \mathrm{~J}\) e) 7360 J

Short answer

Question: A person with mass 75.0 kg climbs a flight of stairs at constant speed reaching a height of 10.0 m. Calculate the work done by the person. Choose the closest value from the given options: a) \(7.36 \cdot 10^{5} \mathrm{~J}\) b) 750 J c) 75 J d) \(7500 \mathrm{~J}\) e) 7360 J

Step-by-step solution

Calculate the force due to gravity

The gravitational force acting on the person (F_gravity) can be calculated using the formula: \(F_{gravity} = m \cdot g\) where \(m\) is the mass of the person and \(g\) is the gravitational acceleration (approximately \(9.81\,\mathrm{m/sec^2}\)).

Compute the gravitational force

Now, using the given mass and gravitational acceleration, let's compute the force due to gravity: \(F_{gravity} = 75.0\,\mathrm{kg} \cdot 9.81\,\mathrm{m/sec^2} = 735.75\,\mathrm{N}\)

Calculate the work done

To calculate the work done by the person against gravity, we can use the formula: Work Done = Force × Distance The force acting on the person is the gravitational force, and the distance is the height of the flight of stairs. So, Work Done = \(F_{gravity} \cdot h = 735.75\,\mathrm{N} \cdot 10.0\,\mathrm{m} = 7357.5\,\mathrm{J}\)

Find the correct answer

Now, we can match the work done with the given options to find the closest value: a) \(7.36 \cdot 10^{5} \mathrm{~J}\) b) 750 J c) 75 J d) \(7500 \mathrm{~J}\) e) 7360 J The closest value to our calculated work done (7357.5 J) is 7360 J, which corresponds to option e. Hence, the correct answer is e) 7360 J.

Key concepts

Gravitational Force

In physics, gravitational force is an attraction that acts between any two objects with mass. According to Newton's law of universal gravitation:

The gravitational force (\( F_{gravity} \)) between two bodies can be calculated using the formula:
\[\begin{equation}F_{gravity} = m \times g\right).\right\right,\quad \right)\quad \quad \quad \end{equation}\]However, the gravitational force also depends on the distance between the two objects, this force decreases as the distance between them increases. In our case of a person climbing stairs, the distance (\( h \text)) is the height they ascend, and the mass (\( m \text).)This example highlights the role gravity plays in the work done. When the person moves upward, they exert force to overcome gravity. Thus, the work done against gravity is the product of the force due to gravity and the vertical distance moved.To make this concept more graspable for students, real-world examples like lifting an object upwards or the feeling of weight while climbing stairs can be useful illustrations of gravitational force at work.

Kinetic Energy

The term kinetic energy refers to the energy that an object possesses due to its motion. It is given by the formula:

\[\begin{equation}KE = \frac{1}{2} m v^2,\right).\right\right,\quad \right)\quad \quad \quad \end{equation}\]Where 'KE' represents the kinetic energy, 'm' is the mass of the object, and 'v' is its velocity. In our textbook exercise, the person is climbing the stairs at a constant speed, which implies that their kinetic energy remains the same during the climb.

Despite kinetic energy not being the direct subject of work calculation in this problem, understanding that moving objects have kinetic energy can be crucial. When climbing stairs, if the person were to accelerate, they would also be doing work to increase their kinetic energy. This ties back to the broader principle of energy transformation and conservation.

Conservation of Energy

The conservation of energy principle is a fundamental concept in physics which states that energy cannot be created or destroyed in an isolated system; it can only be transformed from one form to another.

When a person climbs stairs, the chemical energy from the person's muscles is converted into gravitational potential energy as they rise against the pull of gravity. The energy can also transform into kinetic energy if the person speeds up or slows down. If they climb at a constant speed, as in our exercise example, there is no change in kinetic energy, and all of the work done against gravity converts to potential energy.

To provide students with a vivid illustration: it's like charging a battery (potential energy) by pedaling a stationary bike (doing work). When the climber reaches the top of the stairs, the 'battery' is fully 'charged,' ready to be 'spent' on potential energy when they descend. Grasping this concept aids students in understanding various energy transformations in day-to-day activities and the natural world.