Question

Calculate A farmhand pushes a bale of hay 3.9 m across the floor of a barn. She exerts a force of 88 N at an angle of 25 below the horizontal. How much work has she done?

Short answer

The work done is approximately 310.94 Joules.

Step-by-step solution

Understanding Work

Work is defined as the product of the force applied and the displacement over which the force is applied. It can be calculated using the formula: \[ W = F \cdot d \cdot \cos(\theta) \]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 direction of displacement.

Identifying Given Values

From the problem, we know:- The force \( F = 88 \text{ N} \)- The displacement \( d = 3.9 \text{ m} \) - The angle \( \theta = 25^\circ \) below the horizontal.

Calculating Work Done

Using the formula for work, substitute the known values:\[ W = 88 \cdot 3.9 \cdot \cos(25^\circ) \]First, calculate \( \cos(25^\circ) \) which is approximately 0.906. Then multiply these values together:\[ W = 88 \cdot 3.9 \cdot 0.906 \approx 310.940 \, \text{Joules} \]

Key concepts

Force

In physics, force is a fundamental concept that describes an interaction that results in a change in an object's motion. When we talk about force, we often describe it in terms of magnitude and direction. This means that force is a vector quantity, which distinguishes it from scalar quantities.

• **Magnitude**: The size or quantity of the force, usually measured in Newtons (N).
• **Direction**: The direction in which the force is applied, crucial for calculating the actual "effectiveness" of the force when it relates to work.

When our farmhand pushes the bale of hay, she exerts a force of 88 Newtons. This force acts to move the hay across a specified distance or displacement, and part of understanding the work done involves understanding how the force vector's direction plays into this.

Displacement

Displacement in the context of physics refers to how far an object moves from its initial position, measured in a straight line in a specific direction. Unlike distance, which is simply how much ground an object covers, displacement considers direction as well and is also a vector quantity.

In our example, the bale of hay is moved 3.9 meters. This is considered the displacement of the hay. To calculate work, we need to see how this displacement in space interacts with the force being applied. The displacement vector, like force, must be carefully considered in terms of both magnitude and direction for accurate calculations.

Angle

The angle in physics, especially in the context of work,” describes the orientation between two vectors: the force vector and the displacement vector. Sometimes force is not applied directly in line with the displacement, and understanding the angle between these two factors is vital for calculating work.

In the case of the farmhand, the force is applied at a 25-degree angle below the horizontal. This angle is crucial because it affects how much of the force is actually acting in the direction of the displacement, which is essential for calculating the work done.

• **Horizontal Component**: This is the component of the force that works along the direction of the displacement.
• **Vertical Component**: Although present, this component doesn’t contribute to the work, since work considers force in the direction of movement.

Adjustments for the angle are made using trigonometric functions, assisting us in isolating the part of the force that effectively performs work.

Cosine Function

The cosine function, a fundamental trigonometric function, becomes very handy in physics calculations involving angles. When calculating work, especially with angles involved, the cosine function helps us determine the effective component of the force in the direction of displacement.

The formula for work is \[ W = F \cdot d \cdot \cos(\theta) \] In this equation, \( \cos(\theta) \) extracts the horizontal component of the force when the force is applied at an angle. Using the cosine function helps us mathematically break down the problem:

• If \( \theta = 0 \), cos(0) = 1, the full force is in the direction of displacement.
• If \( \theta = 90 \), cos(90) = 0, no work is done as the force is perpendicular to displacement.

In the exercise, calculating \( \cos(25^\circ) \) gives us approximately 0.906. This informs us that a significant portion of the 88 N force contributes to moving the hay forward, allowing us to accurately solve the work done.