Question

Calculate the work done in the following situations. A suitcase is pulled \(50 \mathrm{ft}\) along a horizontal sidewalk with a constant force of 30 ib at an angle of \(30^{\circ}\) above the horizontal.

Short answer

Answer: The work done in pulling the suitcase along the horizontal sidewalk is approximately 1299 ft-lb.

Step-by-step solution

Identify the force component in the direction of displacement

Since the force is acting at an angle, we need to decompose it into its horizontal and vertical components. The horizontal component of the force will be used to find the work done, as this is the component acting in the direction of displacement. To find the horizontal component of the force, we can use the trigonometric function cosine as follows: F_horizontal = F * cos(angle), where F is the force and angle is the angle between the force and the horizontal direction.

Find the angle in radians

Before substituting values, convert the given angle from degrees to radians as follows: angle (radians) = (angle (degrees) × π) / 180 Substitute the given values: angle (radians) = (30 × π) / 180 = π / 6

Calculate the horizontal component of the force

Use the horizontal force equation and the angle in radians to find the horizontal component of the force: F_horizontal = F * cos(angle) F_horizontal = 30 * cos(π / 6) Calculate the value: F_horizontal ≈ 25.98 lb

Calculate the work done

To find the work done, use the formula: W = F_horizontal * d, where W is the work done and d is the displacement. Substitute the given values and horizontal force component: W = 25.98 * 50 Calculate the value: W ≈ 1299 ft-lb The work done in pulling the suitcase along the horizontal sidewalk is approximately 1299 ft-lb.

Key concepts

Horizontal Component of Force

When a force is applied at an angle to the horizontal direction, it is beneficial to break it down into horizontal and vertical components. This decomposition allows us to focus on the part of the force that actually contributes to the work along the path of motion. To find the horizontal component of a force, you can use the cosine of the angle at which the force is applied, since cosine relates the adjacent side (horizontal component) to the hypotenuse (total force) in a right triangle. This formula is expressed as: \[ F_{horizontal} = F \times \cos(\theta) \] Here, \( F \) is the total applied force, and \( \theta \) is the angle. By using this formula, you only consider the effective force that aligns with the direction of travel, enhancing accuracy when calculating work done.

Trigonometry in Physics

Trigonometry is a cornerstone of physics because it helps to break forces into components that line up neatly with the axes of a coordinate system. This is crucial when dealing with non-parallel angles. In our example, the force applied on the suitcase forms a 30° angle with the sidewalk, so we employ trigonometric functions such as cosine to resolve the force into a horizontal component. This component is essential in determining the actual work performed by the force along the horizontal plane.

• **Sine, Cosine, and Tangent:** These functions help relate angles to side lengths in a right triangle.
• **Degrees to Radians:** Physics often involves converting angles from degrees to radians for standardization, using the formula: \[ \theta_{radians} = \frac{\theta_{degrees} \times \pi}{180} \]

Understanding how to apply these concepts allows you to dissect and analyze forces effectively.

Displacement and Force

In physics, both displacement and force are vector quantities, which means they have both magnitude and direction. Work, meanwhile, is a measure of energy transfer that occurs when an object is moved by a force over a distance. In the case of our suitcase scenario, despite the applied force having both a horizontal and vertical component, only the horizontal component aligns with the direction of displacement, and thus contributes to the work done. The overall work done can be calculated using the formula: \[ W = F_{horizontal} \times d \] Here, \( W \) is the work done, \( F_{horizontal} \) is the effective horizontal force, and \( d \) is the displacement. This equation shows why understanding the interaction between displacement and force is key: work is only done by the component of the force that acts along the direction of displacement.