Problem 11 - Applied Physics

Question

An inclined plane is \(10.0 \mathrm{~m}\) long and \(2.50 \mathrm{~m}\) high. (a) Find its mechanical advantage. (b) A resistance of \(727 \mathrm{~N}\) is pushed up the plane. What effort is needed? (c) An effort of \(20 \overline{0} \mathrm{~N}\) is applied to push an \(815-\mathrm{N}\) resistance up the inclined plane. Is the effort enough?

Step-by-Step Solution

Answer

(a) MA = 4.0, (b) Effort = 182 N, (c) Effort is not enough.

Step 1: Understanding Mechanical Advantage

Mechanical advantage (MA) is the ratio of the force exerted by a machine to the force applied to it. For an inclined plane, it is calculated as the ratio of the length of the incline to its height. Mathematically, \( MA = \frac{\text{Length of incline}}{\text{Height of incline}} \). Thus, \( MA = \frac{10.0 \, \text{m}}{2.50 \, \text{m}} \approx 4.0 \).

Step 2: Calculating the Effort Needed (Part B)

To find the effort (E) needed to push a resistance (R) up the plane, use the formula \( E = \frac{R}{MA} \). Here, \( R = 727 \, \text{N} \) and \( MA = 4.0 \). So, \( E = \frac{727}{4.0} \approx 182 \, \text{N} \).

Step 3: Evaluating If the Effort is Enough (Part C)

We need to check if the given effort (200 N) is sufficient to move an 815 N resistance up the plane. Using the formula \( MA = \frac{R}{E} \), the actual required effort is \( E = \frac{815}{4.0} \approx 203.75 \, \text{N} \). Since 200 N is less than 203.75 N, the effort is not enough.

Key Concepts

Inclined Plane

An inclined plane is a simple machine that makes it easier to move an object to a higher elevation using less effort. The idea is quite straightforward: instead of lifting an object straight up, which requires a lot of force, you can slide it up a ramp. This reduces the amount of force needed because the inclined plane extends the distance over which the force is applied.

In this exercise, we have a plane that is 10 meters long and 2.5 meters high. The mechanical advantage of an inclined plane is calculated by taking the length of the slope and dividing it by the height. This gives us a mechanical advantage of 4.0, meaning that the force required to move an object is reduced by four times compared to lifting it vertically.

Some important points about inclined planes include:

• They do not reduce the total amount of work needed, but spread it over a longer distance.
• A higher mechanical advantage means less force is needed.
• Inclined planes do not make moving things easier by reducing the work — they do so by reducing the force needed.

Force Calculation

Force calculation on an inclined plane is essential for understanding how much effort is required to move an object up the slope. Suppose you have an inclined plane with a known mechanical advantage — once you have this, calculating the necessary force becomes a breeze!

The formula to calculate the required effort (E) given a resistance (R) is: \[ E = \frac{R}{MA} \]where \( MA \) is the mechanical advantage. In the provided problem, we found that the mechanical advantage was 4.0. Given a resistance of 727 N, the calculation shows that you'll need approximately 182 N of effort to move this resistance up the plane.

Key takeaways related to force calculation include:

• The larger the mechanical advantage, the less force is required.
• The formula connects the effort, resistance, and mechanical advantage in a clear mathematical relationship.
• Force adjustments are based directly on the calculated mechanical advantage.

Physics Problem Solving

Physics problem-solving often involves dissecting a complex scenario into manageable parts. Here, we applied this by using known formulas and relationships to systematically assess the situation.

Given a challenge, like determining if a specific effort can move a resistance up an inclined plane, approaching it step-by-step is crucial. For instance, to check if 200 N of effort could move 815 N, calculate the necessary effort using:\[ E = \frac{R}{MA} \]This helps determine the actual needed force — 203.75 N in our exercise. We found out 200 N was not sufficient.

In physics problem-solving, always remember:

• Break problems into smaller, logical steps.
• Utilize formulas only when you understand their application.
• Analyze the results to ensure they make sense in the given context.
• Always double-check assumptions and calculations.

Through practice, these methods become second nature in tackling physics problems efficiently.