Question

How much work is performed in moving a box up the length of a 10 ft ramp that makes a \(5^{\circ}\) angle with the horizontal, with 50lb of force applied in the direction of the ramp?

Short answer

The work performed is 500 ft-lb.

Step-by-step solution

Understand the concept of work

Work is defined as the force applied to an object times the distance over which it is applied. The formula to calculate work is \( W = F \times d \times \cos(\theta) \), where \( W \) is work, \( F \) is the force applied, \( d \) is the distance over which the force is applied, and \( \theta \) is the angle between the force and the direction of motion.

Identify the given values

From the problem, we have the following values: \( F = 50 \text{ lb} \) (force applied), \( d = 10 \text{ ft} \) (distance along the ramp), and \( \theta = 0^{\circ} \) (since the force is applied along the direction of the ramp).

Calculate the cosine of the angle

Since the force is applied in the direction of the ramp, the angle \( \theta \) between the force direction and the ramp is \( 0^{\circ} \). Therefore, \( \cos(0^{\circ}) = 1 \).

Apply the work formula

Substitute the values into the work formula: \[ W = F \times d \times \cos(\theta) = 50 \times 10 \times 1 = 500 \text{ ft-lb}. \]

Interpret the result

The calculation shows that 500 foot-pounds of work is performed in moving the box up the ramp.

Key concepts

Force and Motion

Understanding the relationship between force and motion is crucial when analyzing physical scenarios like moving a box up a ramp. Force is any interaction that, when unopposed, will change the motion of an object. Imagine you're pushing a shopping cart; the push you exert is a force that causes the cart to move.

Motion, on the other hand, is the change in position of an object over time. When force is applied in a certain direction, it causes an object to move or change its motion. In our exercise, a force of 50 pounds is applied to move a box up a ramp, which results in motion along the length of the ramp.

It's important to highlight that this force must overcome any opposing forces, such as friction or gravitational pull, to achieve the desired motion. The effective use of force in the direction of the ramp ensures that the box moves efficiently up the incline.

Calculus Applications

Calculus plays a significant role in physics problems, particularly when calculating work done on an object. When a force is variable or when dealing with inclined planes, calculus provides the tools to integrate variables and solve complex equations. However, in this exercise, the force applied is constant, simplifying the application of calculus.

In more advanced scenarios, you might encounter situations where the force isn't constant along the path. This requires integrating the force over the distance, taking into account how force changes. Calculus helps in understanding these dynamics and provides a method to precisely calculate the work done under such circumstances.
- For constant force, like in our problem, applying the simpler work formula is sufficient. - For varying forces, calculus allows summation over small sections to find the total work.

By starting with these fundamental concepts, one can appreciate how calculus provides a deeper insight into the intricacies of physics.

Work Formula in Physics

The work formula in physics provides a straightforward way to calculate the work done on an object when a force is applied. The basic formula is:\[ W = F \times d \times \cos(\theta) \]
- Here, \( W \) is the work done, \( F \) is the force applied, \( d \) is the distance over which the force acts, and \( \theta \) is the angle between the force and the direction of motion.

In our exercise, since the box is moved along the ramp direction, the angle \( \theta \) is \(0^{\circ}\), meaning the force is applied directly along the path. This makes the cosine of the angle equal to 1, reducing the formula to a simple multiplication of force and distance, \( W = F \times d \). Thus, the work done equals \(50 \times 10 = 500 \text{ ft-lb}\).

Remember, the work formula can adapt to different scenarios. Cosine of the angle becomes particularly crucial when the force isn't applied in the direct path of movement. Hence, understanding these components helps in comprehensively approaching various physics problems.