Question

Suppose you pull a sled with a rope that makes an angle of \(30.0^{\circ}\) to the horizontal. How much work do you do if you pull with \(25.0 \mathrm{~N}\) of force and the sled moves \(25.0 \mathrm{~m} ?\)

Short answer

Answer: The work done is 541.25 N·m.

Step-by-step solution

Find the horizontal component of the force

To find the horizontal component of the force, we can use the formula \(F_x = F\cos(\theta),\) where \(F_x\) is the horizontal component of the force, and \(\theta\) is the angle between the rope and the horizontal direction. In this case, the angle is \(30.0^{\circ},\) and the force is \(25.0 \mathrm{~N}.\) So, \(F_x = 25.0\mathrm{~N}\times\cos(30.0^{\circ})\)

Compute the horizontal force

Now that we have the formula for the horizontal force, we can compute the horizontal component of the force: \(F_x = 25.0\mathrm{~N}\times\cos(30.0^{\circ}) = 25.0\mathrm{~N}\times\frac{\sqrt{3}}{2} = 21.65 \mathrm{~N}.\)

Calculate the work done using the formula

Now that we have the horizontal component of the force, we can compute the work done using the formula \(W = F_x d\) where \(W\) is the work done, \(F_x\) is the horizontal component of the force, and \(d\) is the distance. In this case, the distance is \(25.0 \mathrm{~m}.\) So, \(W = 21.65\mathrm{~N}\times 25.0\mathrm{~m}\)

Compute the work done

Computing the expression for the work done, we get: \(W = 21.65\mathrm{~N}\times 25.0\mathrm{~m} = 541.25\mathrm{~N\cdot m}.\) So the work done is \(541.25 \mathrm{~N\cdot m}.\)

Key concepts

Work Done Formula

Understanding the concept of work in physics is essential for analyzing how much energy is transferred during an event. The work done formula is a fundamental expression given by

\[ W = F \times d \times \text{cos}(\theta) \
Where \(W\) stands for work done, measured in joules (J); \(F\) represents the magnitude of the force applied, measured in newtons (N); \(d\) indicates the distance over which the force is applied, measured in meters (m); and \(\theta\) is the angle between the force direction and the direction of motion.

This formula implies that only the component of the force that acts in the direction of motion contributes to work done. If the force is perpendicular to the direction of motion, no work is done; the cosine function reflects this relationship. In a practical sense, if you apply a force to move an object a certain distance, the energy you transfer to the object is the work done.

Horizontal Component of Force

When dealing with forces at angles, as in the sled problem, knowing how to decompose a force into its components is crucial. The horizontal component of force is essentially the part of the total force that acts parallel to the horizontal surface. It's responsible for moving an object along the surface.

To calculate the horizontal component, \( F_x \), of a given force \( F \) that acts at an angle \( \theta \) to the horizontal, we use the equation \[ F_x = F \times \text{cos}(\theta) \
By using this equation, we break a slanted force into a more straightforward horizontal force that we can use to determine how effectively it will move an object horizontally. In our example with the sled, the force applied through the rope includes an angle, which means that not all of the applied force will contribute to horizontally moving the sled. Calculating the horizontal component is fundamental to determine the amount of work done.

Cosine Function in Physics

The cosine function is a crucial mathematical tool in physics, particularly when it comes to understanding the relationship between different components of force and motion. In any situation where a force is applied at an angle to the direction of motion (like pulling a sled at an angle to the ground), the cosine function helps us figure out the effective part of that force in the direction we're interested in.

From trigonometry, we know that \( \text{cos}(\theta) \) gives the adjacent side over the hypotenuse in a right-angled triangle, which in physics translates to the ratio of the horizontal force to the applied force when the angle is with respect to the horizontal axis. It is this ratio that tells us what fraction of the total force actually contributes to the work done in moving an object. Without the cosine function, resolving forces at angles and calculating work done would become much more challenging in many physics problems.