Problem 3 - Single Variable Calculus

Question

A particle moves a distance of \(100 \mathrm{ft}\) along a straight line. As it moves, it is acted upon by a constant force of magnitude \(5 \mathrm{lb}\) in a direction opposite to that of the motion. What is the work done by the force?

Step-by-Step Solution

Answer

The work done by the constant force acting opposite to the direction of motion is calculated using the formula \(W = Fd \cdot \cos\theta\), with \(F = 5 \mathrm{lb}\), \(d = 100 \mathrm{ft}\), and \(\theta = \pi\). Plugging in the values, we get \(W = 5 \cdot 100 \cdot (-1) = -500 \mathrm{ft\cdot lb}\). Therefore, the work done by the force is -500 ft-lb, and the negative sign indicates that the force is acting opposite to the direction of displacement.

Step 1: Identify the given data

We are given: 1. Distance moved along a straight line, \(d = 100 \mathrm{ft}\). 2. Constant force magnitude, \(F = 5 \mathrm{lb}\). 3. The direction of force is opposite to the direction of motion.

Step 2: Determine the angle between force and displacement vectors

As the force is acting in the opposite direction of the motion, the angle between the two vectors is 180 degrees. We can represent this in radians as \(\theta = \pi\) radians.

Step 3: Calculate the work done

The work done (\(W\)) by the force is given by the equation: \[W = Fd \cdot \cos\theta\] We can plug in the values we have: \(W = 5 \mathrm{lb} \cdot 100 \mathrm{ft} \cdot \cos(\pi)\) As \(\cos(\pi) = -1\), we can simplify the equation: \(W = 5 \cdot 100 \cdot (-1)\) \(W = -500 \mathrm{ft \cdot lb}\) Therefore, the work done by the force is -500 ft-lb. The negative sign indicates that the force is acting opposite to the direction of displacement.

Key Concepts

Constant Force

In physics, a constant force is a force that does not change in magnitude or direction as an object moves. When analyzing problems in mechanics, it's common to encounter scenarios where a constant force is applied to an object. This simplification allows for straightforward calculations when determining quantities like work, which relies on the force remaining the same throughout the motion. In our example problem, the particle experiences a constant force of 5 lb, which means the force is exerted consistently as the particle travels across the 100 ft distance.

Displacement

Displacement refers to the change in position of an object. Unlike distance, which measures the total path traveled, displacement considers only the initial and final positions, straight-line direction included. In our exercise, the particle's displacement is 100 ft as it moves from its starting point to the ending point along a straight line. Displacement is crucial in calculating work because it gives us the direction and magnitude of movement, helping us understand how the applied force interacts with this motion.

Angle Between Vectors

The angle between vectors is a vital concept when calculating work done by a force. The angle determines how much of the force contributes to the movement in the direction of displacement. This is achieved through the dot product calculation, which includes the cosine of the angle. In the problem, the angle between the force applied and the displacement is 180 degrees, implying they are in opposite directions. Since the cosine of 180 degrees is -1, the work done ends up being negative, indicating the force acts against the motion.

Opposite Direction Force

When a force is in the opposite direction to the motion, it tends to slow down or resist the movement of the object. This scenario is seen in braking or objects moving against friction. In our case, the constant force of 5 lb acts opposite to the particle's displacement, leading to negative work done. The mathematical representation of this is the negative sign in the calculation, which shows that the force extracts energy from the system, resisting the particle's motion as it travels the 100 ft distance.