Question

A couch might take \(100 \mathrm{~N}\) to slide across a floor. If someone slides the couch 4 meters and does it in 8 seconds, how much power did they expend?

Short answer

Answer: The person expended 50 Watts of power.

Step-by-step solution

Calculate the work done

Using the formula Work = Force x Distance, we have Work = 100 N * 4 m. Calculate this product: Work = 100 * 4 = 400 J (Joules, a unit of energy).

Calculate the power expended

Now that we have the work done, we can find the power expended by dividing the work by the time it took to move the couch. Using the formula Power = Work / Time, we have Power = 400 J / 8 s. Calculate this quotient: Power = 400 / 8 = 50 W (Watts, a unit of power). So, the person expended 50 Watts of power while sliding the couch 4 meters in 8 seconds.

Key concepts

Work and Energy

Understanding work and energy is central to grasping the fundamentals of physics, especially when it comes to calculating power. Simply put, work is done when a force moves an object over a distance. In our exercise, the force required to slide a couch across a floor is given as 100 newtons (N), and the couch was moved for a distance of 4 meters (m).

The formula for work is expressed as:
\[ \text{Work} = \text{Force} \times \text{Distance} \]
By performing this multiplication, we find the total work done to be 400 joules (J), where a joule is the standard unit of energy in physics. Energy reflects the capacity to do work, so when work is done, energy is transferred or transformed. Hence, the energy expended by the person to move the couch is 400 J. Energy is involved in all physical processes and is crucial for performing work, which in this case, comes from the person’s muscles exerting the force to move the couch.

Force and Motion

The force and motion concept is vital in explaining how objects move and what causes them to move. Force is a push or pull that can change the motion of an object, as seen with the couch. Motion refers to the continuous change in the position of an object. According to Newton's laws, particularly the second law, the sum of the forces on an object equals the mass times the acceleration (\( F = ma \)).

In this exercise, we are dealing with a static force of 100 N that overcomes the friction between the couch and the floor to produce motion over 4 meters. Although we did not directly calculate acceleration, understanding that a force is required to change the state of motion of the couch helps to decipher the relationship between force and displacement. The ability to consistently apply force over a distance is indicative of the energy expended and work done.

Power Formula in Physics

The final piece of our problem involves the power formula in physics. Power is defined as the rate at which work is done or energy is transferred. It answers the question: how fast is work being done? The faster work is done, the greater the power. Mathematically, power is the work done divided by the time it takes to do it.

Using the formula: \[ \text{Power} = \frac{\text{Work}}{\text{Time}} \]
we determine how much power was expended to move the couch. After substituting in our values (400 J for work and 8 seconds for time), we calculate that the power expended was 50 watts (W). Watts are the unit of power, which is named after James Watt, the inventor who greatly improved the steam engine in the 18th century. This quantitative assessment of power helps inform about not just the total energy utilized, but also the efficiency and speed of the work done, which is invaluable in various applications in physics and engineering alike.