Problem 3 - Calculus of a Single Variable

Question

Constant Force In Exercises \(1-4,\) determine the work done by the constant force. A force of 112 newtons is required to slide a cement block 8 meters in a construction project.

Step-by-Step Solution

Answer

The work done in sliding the cement block is 896 joules.

Step 1: Identify the Force and Displacement Given

In this problem, a force of 112 newtons is required to slide a cement block 8 meters. Therefore, the force (F) = 112 newtons and the displacement (d) = 8 meters,

Step 2: Understand the Concept of Work Done

Work done on an object is a result of the force applied on it over a distance. The formula used to calculate this is \(W = F \cdot d \cdot cos(θ)\), where W is the work done, F is the force, d is the distance and \(θ\) is the angle at which the force is applied. As the force applied is in the same direction as the movement of the object, \(θ = 0\). Consequently, \(cos(θ) = 1\). Thus, our formula simplifies to \(W = F \cdot d\).

Step 3: Calculate the Work Done

Substitute the values of force and displacement into the formula. This gives work = (112 newtons) * (8 meters). Calculating gives the work done as 896 joules.

Key Concepts

Force and Displacement

When understanding work done by a force, it’s essential to know what both "force" and "displacement" mean. A force is basically a push or pull that can cause an object to move. In our scenario, the force applied is 112 newtons. This is the strength of the push needed to move the cement block.
Displacement, on the other hand, is the distance over which the force is applied, measured in meters in our case. We have to move the block a distance of 8 meters.
This relationship between force and the distance over which it acts is crucial for calculating work done. The larger either factor (force or displacement), the more effort required, thus more work is done.

Work Formula

The concept of "work done" in physics is captured by a straightforward formula. It enables us to calculate how much energy is used to move an object. The formula is: \[ W = F \cdot d \cdot cos(θ) \]
Here’s what each symbol stands for:

• \(W\): Work done, measured in joules
• \(F\): Force applied, measured in newtons
• \(d\): Displacement, or the distance the object moves, measured in meters
• \(θ\): Angle between the force applied and the direction of displacement

In situations where the force applied is in the exact direction of movement, the formula simplifies to \(W = F \cdot d\) because \(cos(θ) = 1\). This is what happens in our block-sliding exercise.

Angle of Force Application

The angle at which force is applied relative to the direction of movement significantly affects the work done. In physics, this angle is represented by \(θ\). Why does the angle matter? The force is most effective—hence the highest work is done—when applied directly along the direction of displacement (\(θ = 0\)). In this case, our formula simplifies greatly because \(cos(0) = 1\). This means no energy or work is "lost" in another direction.
When \(θ\) is not zero, part of the force is "wasted" pushing in a direction that doesn’t contribute to moving the object forward. Therefore, either more force is required or less work is done with the same force.