Calculating the Unit Vector

Oct 13, 2025 by wbrum.org 30 views

Introduction

In physics, work is a measure of the energy transferred by a force to an object as it moves through a distance. The work done by a force is calculated using the formula: W = F * d * cos(θ), where W is the work done, F is the magnitude of the force, d is the distance over which the force is applied, and θ is the angle between the force and the direction of motion. In this article, we will use this formula to find the work done by a force of 21 pounds acting in the direction (2, 5) in moving an object 7 feet from (0, 0) to (7, 0).

Understanding the Problem

The problem states that a force of 21 pounds is acting in the direction (2, 5) and is used to move an object 7 feet from (0, 0) to (7, 0). To find the work done by this force, we need to calculate the angle between the force and the direction of motion. We can do this by finding the unit vector in the direction of the force and then taking the dot product of this unit vector with the unit vector in the direction of motion.

Calculating the Unit Vector

The unit vector in the direction of the force can be found by dividing the force vector by its magnitude. The force vector is given by F = (2, 5), so its magnitude is |F| = sqrt(2^2 + 5^2) = sqrt(4 + 25) = sqrt(29). The unit vector in the direction of the force is then:

u_F = (2/sqrt(29), 5/sqrt(29))

Calculating the Unit Vector in the Direction of Motion

The unit vector in the direction of motion can be found by dividing the displacement vector by its magnitude. The displacement vector is given by D = (7, 0), so its magnitude is |D| = sqrt(7^2 + 0^2) = sqrt(49) = 7. The unit vector in the direction of motion is then:

u_D = (7/7, 0/7) = (1, 0)

Finding the Angle Between the Force and the Direction of Motion

The angle between the force and the direction of motion can be found by taking the dot product of the unit vectors in the direction of the force and the direction of motion. The dot product is given by:

u_F · u_D = (2/sqrt(29)) * (1) + (5/sqrt(29)) * (0) = 2/sqrt(29)

The angle between the force and the direction of motion is then:

θ = arccos(2/sqrt(29))

Calculating the Work Done

Now that we have the angle between the force and the direction of motion, we can calculate the work done by the force using the formula: W = F * d * cos(θ). Plugging in the values, we get:

W = 21 * 7 * cos(arccos(2/sqrt(29)))

Using a calculator to evaluate the expression, we get:

W ≈ 136.49 foot-pounds

Conclusion

In this article, we used the formula W = F * d * cos(θ) to find the work done by a force of 21 pounds acting in the direction (2, 5) in moving an object 7 feet from (0, 0) to (7, 0). We calculated the unit vector in the direction of the force and the unit vector in the direction of motion, and then found the angle between the force and the direction of motion. Finally, we used this angle to calculate the work done by the force. The result is approximately 136.49 foot-pounds.

Answer

Introduction

In our previous article, we discussed how to find the work done by a force using the formula W = F * d * cos(θ). In this article, we will answer some common questions related to work done by a force.

Q: What is the difference between work and energy?

A: Work and energy are related but distinct concepts in physics. Work is the transfer of energy from one object to another through a force, while energy is the ability to do work. In other words, work is a measure of the energy transferred, while energy is the capacity to perform work.

Q: What is the unit of work?

A: The unit of work is typically measured in joules (J) in the International System of Units (SI). However, in the United States, work is often measured in foot-pounds (ft-lb).

Q: What is the formula for work done by a force?

A: The formula for work done by a force is W = F * d * cos(θ), where W is the work done, F is the magnitude of the force, d is the distance over which the force is applied, and θ is the angle between the force and the direction of motion.

Q: What is the significance of the angle between the force and the direction of motion?

A: The angle between the force and the direction of motion is crucial in determining the work done by a force. If the force and the direction of motion are perpendicular (θ = 90°), the work done is zero. If the force and the direction of motion are parallel (θ = 0°), the work done is maximum.

Q: Can work be negative?

A: Yes, work can be negative. If the force and the direction of motion are in opposite directions, the work done is negative. This means that the force is doing work against the motion, rather than with it.

Q: What is the relationship between work and kinetic energy?

A: Work and kinetic energy are related through the equation W = ΔKE, where W is the work done and ΔKE is the change in kinetic energy. This means that the work done on an object is equal to the change in its kinetic energy.

Q: Can work be done by a force on an object if the object is not moving?

A: Yes, work can be done by a force on an object even if the object is not moving. This is known as potential energy, which is stored energy that can be converted into kinetic energy when the object is set in motion.

Q: What is the difference between work and torque?

A: Work and torque are related but distinct concepts in physics. Work is the transfer of energy from one object to another through a force, while torque is a measure of the rotational force that causes an object to rotate.

Conclusion

In this article, we answered some common questions related to work done by a force. We discussed the difference between work and energy, the unit of work, the formula for work done by a force, and the significance of the angle between the force and the direction of motion. We also explored the relationship between work and kinetic energy, and the difference between work and torque.