# How to Calculate Work and Energy

Understanding the concepts of work and energy is essential in the field of physics. These concepts are not only theoretical but also have practical applications in everyday life. Work is done when a force is applied to an object and the object moves through a distance as a result of that force. Energy, on the other hand, is a measure of a system’s ability to do work. To calculate work and energy, certain formulas are typically used.

## Work

Work (\(W\)) is defined as the product of the force (\(F\)) applied to an object in the direction of the displacement and the displacement (\(d\)) itself.

\[ W = F \cdot d \cdot \cos(\theta) \]

Here, \(W\) is work, \(F\) is the force applied, \(d\) is the displacement, and \(\theta\) is the angle between the force vector and the displacement vector. In the International System of Units (SI), work is measured in joules (J).

If the force is applied in the same direction as the object’s movement, the formula simplifies to:

\[ W = F \cdot d \]

## Energy

Energy comes in many forms such as kinetic energy, potential energy, thermal energy, etc. Kinetic energy is the energy of motion, and it’s calculated as:

\[ KE = \frac{1}{2} mv^2 \]

where \(KE\) is kinetic energy, \(m\) is mass, and \(v\) is velocity.

Potential energy, on the other hand, can take various forms but is often associated with gravitational potential energy in high school physics, expressed as:

\[ PE = mgh \]

where \(PE\) is potential energy, \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is the height.

Energy is also measured in joules (J) in the SI system.

Now, let’s work on some problems and solutions related to calculating work and energy.

### Problems and Solutions

**Problem 1:** A box is dragged across the floor with a force of 20 N for a distance of 10 meters. Calculate the work done.

**Solution 1:**

\[ W = F \cdot d = 20 \text{ N} \cdot 10 \text{ m} = 200 \text{ J} \]

**Problem 2:** A car weighing 1500 kg is moving with a velocity of 20 m/s. Calculate its kinetic energy.

**Solution 2:**

\[ KE = \frac{1}{2} mv^2 = \frac{1}{2} \cdot 1500 \text{ kg} \cdot (20 \text{ m/s})^2 = 300,000 \text{ J} \]

**Problem 3:** Calculate the gravitational potential energy of a 500 kg object placed at a height of 8 meters above the ground. Take \(g = 9.8 \text{ m/s}^2\).

**Solution 3:**

\[ PE = mgh = 500 \text{ kg} \cdot 9.8 \text{ m/s}^2 \cdot 8 \text{ m} = 39,200 \text{ J} \]

**Problem 4:** A person is pushing a shopping cart with a force of 50 N. If he pushes it for 30 meters, how much work does he do?

**Solution 4:**

\[ W = F \cdot d = 50 \text{ N} \cdot 30 \text{ m} = 1500 \text{ J} \]

**Problem 5:** A batter hits a 0.15 kg baseball, giving it a velocity of 40 m/s. What is the kinetic energy of the baseball?

**Solution 5:**

\[ KE = \frac{1}{2} mv^2 = \frac{1}{2} \cdot 0.15 \text{ kg} \cdot (40 \text{ m/s})^2 = 120 \text{ J} \]

**Problem 6:** A crane lifts a 300 kg beam to the top of a building 100 meters tall. How much work is done by the crane?

**Solution 6:**

\[ W = F \cdot d = mgh = 300 \text{ kg} \cdot 9.8 \text{ m/s}^2 \cdot 100 \text{ m} = 294,000 \text{ J} \]

**Problem 7:** A diver weighing 60 kg jumps from a 10-meter high platform into a pool. What is the potential energy just before the dive?

**Solution 7:**

\[ PE = mgh = 60 \text{ kg} \cdot 9.8 \text{ m/s}^2 \cdot 10 \text{ m} = 5,880 \text{ J} \]

**Problem 8:** A force of 10 N is applied at an angle of 30 degrees to the horizontal to move an object 5 meters. Calculate the work done.

**Solution 8:**

\[ W = F \cdot d \cdot \cos(\theta) = 10 \text{ N} \cdot 5 \text{ m} \cdot \cos(30^\circ) \approx 43.3 \text{ J} \]

**Problem 9:** How much work is done when a 20 N force moves an object 0 meters?

**Solution 9:**

\[ W = F \cdot d = 20 \text{ N} \cdot 0 \text{ m} = 0 \text{ J} \]

(No work is done since there is no displacement.)

**Problem 10:** What is the kinetic energy of a 75 kg sprinter running at 10 m/s?

**Solution 10:**

\[ KE = \frac{1}{2} mv^2 = \frac{1}{2} \cdot 75 \text{ kg} \cdot (10 \text{ m/s})^2 = 3,750 \text{ J} \]

**Problem 11:** A 10 kg object is lifted vertically with a force of 150 N for a distance of 2 meters. Calculate the work done.

**Solution 11:**

\[ W = F \cdot d = 150 \text{ N} \cdot 2 \text{ m} = 300 \text{ J} \]

**Problem 12:** A pendulum of mass 1 kg swings to a height of 0.5 meters. Calculate its potential energy at that height.

**Solution 12:**

\[ PE = mgh = 1 \text{ kg} \cdot 9.8 \text{ m/s}^2 \cdot 0.5 \text{ m} = 4.9 \text{ J} \]

**Problem 13:** A cyclist exerts a force of 200 N to climb a hill of 50 meters. Calculate the work done.

**Solution 13:**

\[ W = F \cdot d = 200 \text{ N} \cdot 50 \text{ m} = 10,000 \text{ J} \]

**Problem 14:** If a motor does 5,000 J of work to move a 100 kg crate across a distance, and the force applied is 200 N, how far was the crate moved?

**Solution 14:**

\[ W = F \cdot d \implies d = \frac{W}{F} = \frac{5,000 \text{ J}}{200 \text{ N}} = 25 \text{ m} \]

**Problem 15:** A machine lifts a 200 kg object up a vertical distance of 1 meter. Calculate the work done assuming no friction.

**Solution 15:**

\[ W = mgh = 200 \text{ kg} \cdot 9.8 \text{ m/s}^2 \cdot 1 \text{ m} = 1,960 \text{ J} \]

**Problem 16:** Calculate the work required to accelerate a car with a mass of 1200 kg from rest to 18 m/s.

**Solution 16:**

First, calculate the kinetic energy at 18 m/s:

\[ KE = \frac{1}{2} mv^2 = \frac{1}{2} \cdot 1200 \text{ kg} \cdot (18 \text{ m/s})^2 = 194,400 \text{ J} \]

The work done is equal to the change in kinetic energy, which in this case is the final kinetic energy since it started from rest:

\[ W = KE = 194,400 \text{ J} \]

**Problem 17:** An elevator motor does 40,000 J of work lifting a 500 kg elevator up 20 meters. Assuming no other forces are acting on the elevator, find the force exerted by the motor.

**Solution 17:**

\[ W = F \cdot d \implies F = \frac{W}{d} = \frac{40,000 \text{ J}}{20 \text{ m}} = 2,000 \text{ N} \]

**Problem 18:** A bow of spring constant 1200 N/m is compressed by 0.05 meters. Calculate the potential energy stored in the spring.

**Solution 18:**

For a spring, the potential energy (elastic potential energy) is given by:

\[ PE = \frac{1}{2}kx^2 \]

where \(k\) is the spring constant and \(x\) is the displacement from its rest position.

\[ PE = \frac{1}{2} \cdot 1200 \text{ N/m} \cdot (0.05 \text{ m})^2 = 1.5 \text{ J} \]

**Problem 19:** A truck does 25,000 J of work in hauling a load across 100 meters. Calculate the average force exerted by the truck.

**Solution 19:**

\[ W = F \cdot d \implies F = \frac{W}{d} = \frac{25,000 \text{ J}}{100 \text{ m}} = 250 \text{ N} \]

**Problem 20:** A child slides down a slide 3 meters high. If the child’s mass is 30 kg, calculate the potential energy at the top and the kinetic energy at the bottom, assuming no energy loss.

**Solution 20:**

Potential energy at the top:

\[ PE_{\text{top}} = mgh = 30 \text{ kg} \cdot 9.8 \text{ m/s}^2 \cdot 3 \text{ m} = 882 \text{ J} \]

Since there’s no energy loss (ideal situation), the potential energy at the top becomes the kinetic energy at the bottom:

\[ KE_{\text{bottom}} = PE_{\text{top}} = 882 \text{ J} \]

Calculating work and energy involves using fundamental formulas and understanding the conditions under which work is done or energy is consumed/stored. These practice problems offer a mix of scenarios to strengthen comprehension and proficiency with these important physics calculations.