Problem 16 - Applied Physics

Question

A plank \(12 \mathrm{ft}\) long is used as an inclined plane to a platform \(3.0\) ft high. (a) What force must be used to push a load weighing 480 lb up the plank? (b) Find the MA of the inclined plane.

Step-by-Step Solution

Answer

The force required is 120 lbs, with a mechanical advantage of 4.

Step 1: Understanding the Inclined Plane

We have a plank that acts as an inclined plane with a length (hypotenuse) of \(12\, \text{ft}\) and a height (vertical leg) of \(3\, \text{ft}\). The load to be pushed weighs \(480\, \text{lbs}\). To approach this, we need to determine the angle of inclination and use it to find the force required.

Step 2: Calculate the Sine of the Angle

The force required to push the load up the plane is determined by the sine of the angle of incline. Calculate \(\sin(\theta)\) using the formula: \(\sin(\theta) = \frac{\text{height}}{\text{length}}\). Thus, \(\sin(\theta) = \frac{3}{12} = \frac{1}{4}\).

Step 3: Calculate the Applied Force

The force needed to push the load up the inclined plane can be computed with the formula: \(F = \text{Weight} \times \sin(\theta)\). Substituting the values, we get: \(F = 480 \times \frac{1}{4} = 120\, \text{lbs}\).

Step 4: Calculate the Mechanical Advantage (MA)

To find the mechanical advantage of the incline, use the formula \(MA = \frac{\text{Length of the Incline}}{\text{Height of the Incline}}\). Substituting the values, \(MA = \frac{12}{3} = 4\).

Key Concepts

Mechanical Advantage

Mechanical Advantage (MA) plays a significant role in understanding inclined planes. Imagine moving a heavy object uphill. The inclined plane allows you to exert less force than lifting it vertically. MA is calculated as the ratio of the length of the inclined plane to the height. This means you compare how far you're actually moving along the incline to how high you're lifting the object.

• For our plank example, MA = \( \frac{12}{3} = 4 \)
• This implies that you're effectively gaining a fourfold force advantage.

This simplifies how much effort you need to use a fraction of the direct lifting force—making movement up the slope easier. Having a higher mechanical advantage means a more efficient inclined plane.

Force Calculation

Calculating the force required to push a weight up an inclined plane is an essential part of physics. This force helps overcome gravity and any friction, depending on the incline's surface. In our scenario, this involves applying a force "up the hill."To find the force needed, multiply the weight of the object by the sine of the inclination angle:

• Weight = 480 lbs
• Result from calculation: Applied Force \( F = 480 \times \frac{1}{4} = 120 \text{ lbs} \)

This equation simplifies the computation and provides a clear path from input (weight) to the necessary applied force.

Inclination Angle

The inclination angle is critical for understanding how effectively an inclined plane can reduce the applied effort. First, it gives perspective on steepness. A steeper angle means more effort, and a gentler one means less.Calculate this angle using the sine formula:

• \( \sin(\theta) = \frac{\text{height}}{\text{length}} \)
• For our plane: \( \sin(\theta) = \frac{3}{12} = \frac{1}{4} \)

This ratio does not give the angle directly. Instead, it contributes to understanding the amount of force and resistance experienced.

Applied Force

The concept of Applied Force is about what you need to exert to move an object along the incline. Understanding this force guides you on how much effort is necessary versus directly lifting.In our inclined plane exercise, the applied force equals the weight of the load modified by the slope's angle:

• Weight: 480 lbs
• Sine of angle: \( \frac{1}{4} \)
• Applied Force computes to \( 120 \text{ lbs} \)

Therefore, the applied force represents a way of managing work efficiently using the incline feature.

Trigonometry in Physics

Trigonometry is often the backbone of calculations in physics, like problems involving inclined planes. Particularly, it helps translate height, length, and types of forces using right-angle triangle properties.For our inclined plane, the sine ratio relates height and hypotenuse:

• Sin(θ) helps calculate essential forces and work.
• Used to derive formulas like \( F = \text{Weight} \times \sin(\theta) \)

Understanding these applications in inclined planes gives clear insights into optimizing force and energy use.