Problem 22 - College Trigonometry

Question

Find the work done lifting a 10 pound book 3 feet straight up into the air. Assume the force of gravity is acting straight downwards.

Step-by-Step Solution

Answer

The work done is 30 foot-pounds.

Step 1: Understanding Work Formula

The formula to calculate work done, when a force is applied over a distance, is given by \( W = F \times d \), where \( W \) is the work done, \( F \) is the force applied, and \( d \) is the distance over which the force is applied.

Step 2: Identify Known Values

From the problem statement, the force \( F \) is the weight of the book, which is 10 pounds (since force due to gravity is weight), and the distance \( d \) the book is lifted is 3 feet.

Step 3: Calculate Work Done

Substitute the known values into the work formula: \( W = 10 \text{ pounds} \times 3 \text{ feet} = 30 \text{ foot-pounds} \).

Step 4: Final Result

The work done in lifting the 10-pound book 3 feet into the air is 30 foot-pounds.

Key Concepts

Force and Motion

In physics, force and motion are inseparable concepts that help us understand how objects move and interact. A force is any interaction that, when unopposed, will change the motion of an object. It can be a push or pull acting upon an object as a result of its interaction with another object. Motion, on the other hand, refers to the change in position of an object with respect to time.

For example, when you lift a book, you apply a force to overcome gravity and move the book. The book's motion from a lower position to a higher one shows how force and motion work in tandem. Understanding force and motion allows us to calculate work, as motion is a key component in determining how much work is done.

Calculation of Work

Calculating work in physics involves determining how much energy is transferred when a force causes an object to move. The fundamental formula for work (W) is given by:

• \( W = F \times d \)

where \( F \) is the force applied and \( d \) is the distance over which the force is applied.

In the example provided, the force is the weight of the book, a direct result of gravity, and the distance is how far the book is lifted. Substituting these values into the formula gives \( W = 10 \text{ pounds} \times 3 \text{ feet} = 30 \text{ foot-pounds} \). This result shows how much energy is used to lift the book against the force of gravity.

Gravity

Gravity is a natural phenomenon by which all things with mass are brought toward one another, including objects ranging from atoms to planets. On Earth, gravity provides the force that gives weight to physical objects and causes them to fall toward the ground when dropped.

When lifting an object vertically, like a 10-pound book, you are working against the force of gravity acting downward. The force due to gravity on Earth is approximately 9.8 m/s², but is often simplified to 32 feet/s² when using imperial units, as in this exercise.

This gravitational pull is what you must overcome by applying a force equal to the weight of the object to lift it, which in this case is measured in pounds force.

Basic Physics Formulas

Understanding the basic physics formulas is crucial for solving problems related to force, motion, and energy. The key formula used in calculating work is just one among many essential equations in physics.

• Newton's Second Law: \( F = m \cdot a \), describing the relationship between an object's mass \( m \), its acceleration \( a \), and the applied force \( F \).
• Law of Universal Gravitation: \( F = G \frac{m_1 \cdot m_2}{r^2} \), explaining the gravitational attraction between two masses.

These formulas provide a foundation for analyzing various physical scenarios. By mastering these equations, students can better understand complex physical interactions and solve real-world problems effectively.