Question

How much work is done when a \(75-\mathrm{kg}\) person climbs a flight of stairs \(10 \mathrm{~m}\) high at constant speed? a) \(7.35 \cdot 10^{5}\) J c) 75 e) 7350 J b) 750 J d) 7500 J

Short answer

a) \(7.35 \cdot 10^{5}\) J c) 75 J e) 7350 J b) 750 J d) 7500 J

Step-by-step solution

Identify the Mass and Height

Given in the problem, we know that the mass of the person (m) is \(75 \mathrm{kg}\), and the height of the stairs (h) is \(10\mathrm{~m}\).

Calculate the Force Exerted

To calculate the force exerted, we use the formula for gravitational force, \(F = mg\). Here, \(g = 9.81 \mathrm{m/s^2}\) is the gravitational acceleration. So, \(F = (75 \mathrm{kg})(9.81 \mathrm{m/s^2})\). Calculate this product to get the force exerted in climbing the stairs.

Calculate Work Done

Now, we know the distance is equal to the height of the stairs, \(d = h = 10\mathrm{~ m}\). We can calculate the work done (W) using the formula \(W = Fd\): \(W = F \times d\). Using the values obtained in Step 2 and the height of the stairs, calculate the work done.

Choose the Correct Answer

Compare the calculated work done value from Step 3 to the options given in the exercise: a) \(7.35 \cdot 10^{5}\) J c) 75 J e) 7350 J b) 750 J d) 7500 J Determine which option matches the calculated value for the work done while climbing the stairs.

Key concepts

Work Done

When we talk about work done in physics, we're referring to how much energy is transferred when an object is moved over a distance by a force. In our exercise, the person is climbing stairs, and the work done is the energy required to lift their body up to a certain height. This is calculated using the formula:

• The formula is: \[ W = F \times d \] where
  • \(W\) is the work done,
  • \(F\) is the force exerted,
  • \(d\) is the distance over which the force is applied, which is the height of the stairs in this case.

For our problem, once we know the force (calculated through gravitational force), multiply it by the stair height to get the work done. This measurement is in Joules (J), which is the standard unit of work or energy in physics. Understanding this process helps in identifying how energy is used in everyday actions like walking up stairs.

Gravitational Force

Gravitational force is the force of attraction between two masses. On Earth, this is the force that keeps us grounded and impacts how we calculate several physics problems, including our stair-climbing example. For the person climbing the stairs, gravitational force is crucial for determining the force required to do the work.To calculate the gravitational force (\[ F = mg \]), we multiply:

• \(m\) — the mass of the person, here it's 75 kg.
• \(g\) — the acceleration due to gravity, approx. 9.81 m/s² on Earth.

This formula provides us the force exerted by gravity on the person, translating into the force needed to lift them up the stairs. By understanding these calculations, the mystery behind gravity’s role in work and energy becomes a little clearer.

Calculating Force

When calculating force, we begin with identifying the mass and the acceleration due to gravity acting on it. As seen in our exercise, using the gravitational force formula is essential: \\(F = mg\). Here are the necessary steps to calculate force:

• Find the mass of the object or person involved; in this case, it's 75 kg.
• Use the standard gravity value of 9.81 m/s² for Earth's surface.
• Multiply these values to find the force: \\(F = 75 \,\text{kg} \times 9.81 \,\text{m/s}^2\).

Calculating force is a fundamental element in many mechanics problems, particularly when analyzing movement and the support required to move up, as shown in stair climbing. With this understanding, solving for forces in various contexts becomes a straightforward puzzle.

Kinematics

Kinematics is the branch of physics that deals with motion without considering the forces that cause it. However, in our staircase problem, understanding the motion is just as important as calculating the forces involved. By recognizing the uniform motion—climbing the stairs at a constant speed—we simplify our calculations since there's no change in velocity impacting the computations. This uniformity allows using direct measurements without adjustments for acceleration. In essence, kinematics helps us describe motion clearly, laying the groundwork for complex dynamics without initial complicacies about motion-causing forces. When combined with force calculations, it strengthens our grasp of mechanical behavior in real-world scenarios like climbing stairs.