Question

A farmhand pushes a bale of hay \(3.9 \mathrm{~m}\) across the floor of a barn. She exerts a force of \(88 \mathrm{~N}\) at an angle of \(25^{\circ}\) below the horizontal. How much work has she done?

Short answer

The work done by the farmhand is approximately 311.13 J.

Step-by-step solution

Understand the Problem

We need to determine the work done by the farmhand in pushing the bale of hay. The work done can be calculated if we know the force exerted, the displacement, and the angle between the force and the displacement.

Recall the Work Formula

The formula for work done, when a force is applied at an angle, is given by the equation: \[ W = F imes d imes ext{cos} \theta \] where \( W \) is the work done, \( F \) is the force applied, \( d \) is the displacement, and \( \theta \) is the angle between the force and the horizontal direction.

List the Given Values

From the problem, we know: - Force \( F = 88 \mathrm{~N} \)- Displacement \( d = 3.9 \mathrm{~m} \)- Angle \( \theta = 25^{\circ} \) below the horizontal.

Substitute Values into the Work Formula

Substitute the given values into the equation:\[ W = 88 \mathrm{~N} \times 3.9 \mathrm{~m} \times \cos(25^{\circ}) \]

Calculate the Cosine of the Angle

First, calculate \( \cos(25^{\circ}) \) which is approximately 0.9063.

Calculate the Work Done

Substitute the calculated cosine value into the equation:\[ W = 88 \times 3.9 \times 0.9063 \] This gives \[ W \approx 311.12932 \mathrm{~J} \].

Round the Answer

Round the work done to a reasonable number of significant figures, typically two or three based on the precision of given data. Thus, the work done is approximately \( 311.13 \mathrm{~J} \).

Key concepts

Force and Displacement

Understanding how force and displacement interact is fundamental in calculating work done in physics. Force is any interaction that, when unopposed, changes the motion of an object. Displacement, on the other hand, refers to the distance over which the force is applied and is a vector quantity, meaning it has both magnitude and direction.
When a force is applied to an object, causing it to move, the direction and magnitude of both force and displacement need to be taken into account.
This is because work is only done when there is movement in the direction of the force. For example, if you push a box across the floor by applying a force, the displacement would be the distance the box moves along the floor.

• In the case of moving a bale of hay, the farmhand applied a 88 N force and caused a 3.9 m displacement.
• Both factors are crucial in calculating the work done.

Angle in Work Calculation

The angle at which force is applied relative to the direction of displacement plays a crucial role in calculating work. When force is applied at an angle, only the component of the force that is in the direction of the displacement contributes to the work done. To understand this better, picture an angle above or below the horizontal line; only part of the force is effective in moving the object horizontally.

The formula for calculating work done incorporates this angle, given by \( W = F \times d \times \cos \theta \). Here, \( \theta \) is the angle between the force exerted and the direction of displacement.

• In the example, the force is applied at a 25-degree angle below the horizontal.
• This angle affects how much of the force's energy contributes to moving the hay horizontally.

Cosine Function in Physics

The cosine function is used in physics to handle forces applied at an angle effectively. In the formula for work, \( \cos(\theta) \) adjusts the force measurement to account for only the component that moves with the object’s displacement.
This mathematical function, depends on the angle between the force direction and the displacement direction.
To calculate, you take the cosine of the angle, often found using a calculator or trigonometry tables.

• For instance, \( \cos(25^\circ) \approx 0.9063 \).
• The value of the cosine diminishes as the angle increases, reflecting a smaller force component in the direction of movement.

By knowing \( \cos(\theta) \), you can correctly compute how much of the applied force is effective in doing the work of displacing the object.