Problem 77 - Precalculus with Limits

Question

WORK Determine the work done by a person lifting a 245-newton bag of sugar 3 meters.

Step-by-Step Solution

Answer

The work done by a person lifting a 245-newton bag of sugar 3 meters is 735 Joules.

Step 1: Identify the given values

From the problem, it is given that the force (F) or the weight of the bag is 245 newtons (N) and the distance (d) or the height the bag is lifted is 3 meters (m).

Step 2: Use the formula for Work done

The formula for work (W) done is: W = F * d. This formula says that work is the product of the force applied on an object and the distance over which it's applied.

Step 3: Substitute the given values into the formula

Substitute the given values into the formula: W = 245 N * 3 m.

Step 4: Calculate the Work done

Now calculate the product of the force and distance to get the work done: W = 735 N * m. Note: Newton meter (N * m) is the standard unit of work, also known as Joule.

Key Concepts

Force

In physics, "force" is an essential concept that relates to an interaction which causes an object to move. Force is measured in newtons (N). It's the push or pull applied on an object, leading to a change in its state of motion.
To easily understand force, consider these points:

• Force is a vector quantity, which means it has both magnitude and direction.
• Force causes objects to accelerate, decelerate, remain in place, or change direction.
• Common examples of force include gravity, friction, and applied force as seen in our exercise where the bag of sugar is lifted.

In our exercise, the upward force exerted by the person is exactly equal to the weight of the bag of sugar, which is 245 N. This means the force counteracts gravity, allowing the object—a bag in this case—to be lifted from one point to another.

Distance

Distance is a key part of understanding work and energy, as it helps form the basic calculation of work done. Defined as the space between two points, distance is measured in meters (m) in the International System of Units.
In physical science:

• Distance is a scalar quantity, meaning it only has magnitude and no direction.
• It's an important factor in calculating work, as the amount of work depends on how far a force acts upon an object.

In the problem, the distance (3 meters) refers to how high the bag is lifted against gravity. It is crucial to note that the force needs to act in the direction of movement. Thus, for work to be done, the movement (distance) aligns with the direction of the force applied.

Joule

The Joule is the unit of energy or work in the International System of Units. It's a fundamental benchmark in physics for any energy-related calculations. One Joule (J) is equal to one Newton-meter (N*m), which signifies the work done when a force of one newton displaces an object by one meter in the direction of the force.
Exploring the concept of a Joule:

• Joule is a derived unit, combining units of force (newton) and distance (meter).
• It quantifies energy transferred or work done and is crucial in assessing energy efficiency.
• The calculation of work given by \( W = F \times d \) directly leads to a unit measurement in Joules when force is in newtons and distance in meters.

In the exercise, we calculated the work done in lifting the bag as 735 N*m, or 735 Joules, demonstrating how much energy the person spent to lift the bag against the force of gravity.