Problem 16 - Applied Physics

Question

A man pulls a sled a distance of \(231 \mathrm{~m}\). The rope attached to the sled makes an angle of \(30.0^{\circ}\) with the ground. The man exerts a force of \(775 \mathrm{~N}\) on the rope. How much work does the man do in pulling the sled?

Step-by-Step Solution

Answer

The man does approximately 154,563 J of work.

Step 1: Understand the Work Formula

To find the work done by the man, we use the formula for work: \( W = F \cdot d \cdot \cos(\theta) \), where \( W \) is the work done, \( F \) is the force applied, \( d \) is the distance, and \( \theta \) is the angle between the force and the direction of motion.

Step 2: Identify Given Values

From the problem, we have the force \( F = 775 \mathrm{~N} \), the distance \( d = 231 \mathrm{~m} \), and the angle \( \theta = 30.0^{\circ} \). These are the values that need to be substituted into our formula.

Step 3: Substitute and Calculate

Substitute the known values into the work formula: \( W = 775 \cdot 231 \cdot \cos(30.0^{\circ}) \). Calculate \( \cos(30.0^{\circ}) \), which is approximately \( 0.866 \). So, \( W = 775 \cdot 231 \cdot 0.866 \).

Step 4: Perform the Calculation

Calculate the product: \( W = 775 \cdot 231 \cdot 0.866 = 154562.85 \mathrm{~J} \).

Key Concepts

work formula in physics

In physics, work is a concept that involves a force causing an object to move a certain distance. The formula used to calculate work is \( W = F \cdot d \cdot \cos(\theta) \), where:

• \( W \) is the work done, measured in joules (J).
• \( F \) is the force applied, measured in newtons (N).
• \( d \) is the distance over which the force is applied, measured in meters (m).
• \( \theta \) is the angle between the direction of the force and the direction of motion, measured in degrees.

This equation takes into account the fact that only the component of the force that acts in the direction of the movement does the work. Hence, the \( \cos(\theta) \) factor adjusts the force accordingly. If you think about a sled being pulled, it helps us determine how much of the force is directly working to move the sled along the ground.

force and motion

Force and motion are fundamental concepts in physics that describe how objects interact. A force is any interaction that, when unopposed, changes the motion of an object. According to Newton’s laws of motion, forces affect the motion of an object by changing its speed or direction.
When a person pulls a sled, as in our example, a force is applied via the rope, causing the sled to move. The rope's direction is crucial since it's attached at an angle, which impacts the motion. This angle affects how much of the force actually contributes to moving the sled forward along the surface.
To determine the impact of the force on motion, we consider both the magnitude of the force and the angle. This way, we can effectively compute the work done in terms of the sled moving the given distance.

trigonometry in physics

Trigonometry plays a crucial role in physics, especially when it comes to analyzing forces that do not act directly along a line of motion. In our sled example, the rope forms an angle of \( 30.0^{\circ} \) with the horizontal. This is where trigonometry helps by allowing us to break forces into components.

• We use \( \cos(\theta) \) to find the component of the force along the direction of motion.
• \( \sin(\theta) \) can help find perpendicular components, though not required in calculating work directly for our sled example.

The cosine function is especially important when calculating work because it determines the fraction of the total force that actually works to move the sled across the surface. In this scenario, \( \cos(30.0^{\circ}) \) is approximately \( 0.866 \), indicating how much of the applied force contributes to the forward motion across the path.