Problem 221 - Calculus Volume 1

Question

For the following exercises, find the work done. Find the work done when you push a box along the floor \(2 \mathrm{~m}\), when you apply a constant force of \(F=100 \mathrm{~N}\).

Step-by-Step Solution

Answer

200 Joules.

Step 1: Identify the Formula

To calculate the work done, we use the formula: \[W = F \cdot d \cdot \cos(\theta)\]where:- \(W\) is the work done,- \(F\) is the force applied,- \(d\) is the distance over which the force is applied,- \(\theta\) is the angle between the force and the direction of motion. Since the force is applied along the direction of motion, \(\theta = 0\) degrees.

Step 2: Substitute Known Values

Substitute the known values into the formula:\[F = 100 \text{ N}, \quad d = 2 \text{ m}, \quad \theta = 0\]The cosine of 0 degrees is 1, so the formula simplifies to:\[W = 100 \cdot 2 \cdot \cos(0) = 100 \cdot 2 \cdot 1\]

Step 3: Calculate the Work

Perform the multiplication to find the work:\[W = 100 \cdot 2 = 200 \text{ Joules}\]Therefore, the work done in pushing the box is 200 Joules.

Key Concepts

Force and Motion

Force and motion are fundamental concepts in physics. Force is any interaction that, when unopposed, will change the motion of an object. Every force has a magnitude, often measured in Newtons (N), and a direction. Motion refers to the change in the position of an object over time. When force is applied to an object and it causes the object to move, this interaction is often governed by Newton's Laws of Motion. In our exercise, a force of 100 N is applied to push a box. This force causes the box to move a distance of 2 meters. The interaction of force and motion is critical in understanding not just the problem at hand but also in analyzing mechanical interactions in various real-world scenarios. The force acts in the direction of the motion, which means the angle between the direction of the force and the direction of motion is zero degrees. This simplifies our calculation of work to just the product of force and distance.

Work Formula

The work formula is a crucial tool for determining the amount of work done on an object. The formula is \[ W = F \cdot d \cdot \cos(\theta) \]where:

• \( W \) is the work done, measured in joules (J).
• \( F \) is the force applied, measured in newtons (N).
• \( d \) is the distance over which the force is applied, measured in meters (m).
• \( \theta \) is the angle between the force and the direction of motion.

When the force acts in the same direction as the movement, like in our example, \( \theta = 0 \), and \( \cos(0) = 1 \). Thus, the work done simplifies to the product of force and distance: \[ W = F \cdot d \]The work formula helps quantify the energy transferred when a force moves an object, aiding in a deeper understanding of energy dynamics.

Physics Problem Solving

Solving physics problems requires a systematic approach. Here's a simple way to tackle these problems effectively:1. **Identify Key Information**: Start by identifying the quantities given in the problem, like force, distance, or angles.2. **Apply the Relevant Formula**: Use the formula that links these quantities, such as the work formula, to relate force and motion to work done.3. **Simplify the Calculation**: Check for special conditions. In this case, since the force direction matches the motion, simplify the calculation by noting \( \theta = 0 \), which makes \( \cos(\theta) = 1\).4. **Perform the Calculation**: Substitute the given values into the formula and correctly follow through with the arithmetic to solve for the unknown, in this case, work.5. **Verify Your Solution**: Re-check numbers and operations to ensure accuracy.By following these steps, you can confidently solve a wide variety of physics problems, turning challenging scenarios into manageable solutions.