Problem 85 - Precalculus with Calculus Previews

Question

A sled is pulled horizontally over ice by a rope attached to its front. A 20-lb force acting at an angle of \(60^{\circ}\) with the horizontal moves the sled \(100 \mathrm{ft}\). Find the work done.

Step-by-Step Solution

Answer

The work done is 1000 ft-lb.

Step 1: Understand the Work Formula

Work is calculated using the formula: \( W = F \cdot d \cdot \cos(\theta) \), where \( W \) is the work done, \( F \) is the force applied, \( d \) is the distance moved, and \( \theta \) is the angle between the force and the direction of motion.

Step 2: Assign Values to Variables

The force \( F \) is \(20 \text{ lb} \), the distance \( d \) is \(100 \text{ ft} \), and the angle \( \theta \) is \(60^{\circ}\).

Step 3: Use Cosine for the Angle

Calculate the cosine of the angle. Since \( \theta = 60^{\circ} \), we have \( \cos(60^{\circ}) = 0.5 \).

Step 4: Substitute and Calculate Work

Substitute the known values into the work formula: \( W = 20 \cdot 100 \cdot 0.5 \). This simplifies to \( W = 1000 \).

Step 5: State the Unit of Work

Since the force was given in pounds and the distance in feet, the unit of the work done is foot-pounds (ft-lb).

Key Concepts

Force

Force is a fundamental concept in physics, and it plays a crucial role when calculating work and energy. A force is a push or a pull exerted on an object that can cause it to move, stop, or change direction. In our example of the sled, the force is applied via the rope attached to the sled, which pulls it across the ice. This force, measured in pounds (lb), is the amount of energy transferred through an object's interaction with another object.

There are essential characteristics of forces to remember:

• Magnitude: The size or amount of the force applied. In this case, it's a 20-lb force.
• Direction: The line of action along which the force is applied. Here, the force is applied at a 60-degree angle to the horizontal.
• Resultant Force: The total force resulting from combining multiple forces if they act simultaneously.

Understanding force and the way it works helps us know how energy is utilized when moving objects like the sled in the exercise. The force's magnitude and direction directly influence how much work can be done.

Distance

Distance is another key factor in understanding work in physics. It refers to how far an object moves as a result of the force applied. When calculating work, distance is a crucial component since it determines how much energy is used in moving an object from one point to another.

In the sled exercise:

• The distance over which the sled moves is 100 feet.
• It's essential to realize that the distance in this context refers to the path traveled in the direction of the applied force.
• Even if the force is exerted, if the object does not move, then no work is done.

Understanding how distance impacts work emphasizes that without movement along some path or trajectory, no work could be calculated or observed. Thus, distance goes hand in hand with force when discussing the concept of work.

Angle

The angle between the force and the direction of motion is a pivotal concept when calculating work. This angle impacts how much of the force actually contributes to moving the object. The angle can change the effectiveness of the applied force due to how it aligns with the movement direction.

In the sled example:

• The force is applied at a 60-degree angle to the horizontal.
• The cosine of this angle (\( heta = 60^{\circ} \)) influences the calculation by modifying the effective force component in the direction of the sled's motion.
• Using the formula component \( \cos(\theta) \), you can understand how the force is distributed.

Calculating the cosine of the angle helps determine how much of the total force contributes to moving the sled forward. This is why it is included in the work formula, allowing us to compute the real work done by the force parallel to the movement.