Inclined Plane

Inclined Plane as a Simple Machine

The inclined plane is often categorized as one of the simple machines. Sometimes the wedge is classified as a special type of inclined plane, while a minority of texts treat it as its own simple machine. Similarly, the screw is usually defined as an inclined plane that is twisted about a cylinder, rather than its own separate machine.

Acceleration Due to Gravity

An experiment utilizing the inclined plane can be used to verify the value for the acceleration due to gravity, g. To start, a board of a fixed length is positioned various heights above the ground. The time it takes for a ball to roll the length of the board is recorded, along with the vertical height the top of the board is above the ground. Alternatively, the angle the board makes with the ground could be measured. Acceleration due to gravity can then be calculated in one of two ways:

In the above equations, H represents the vertical height of the board above the ground, while L represents the length of the board. Since the ball will be rolled down the board at various heights, H will constantly change, but L (the length of the board and the distance the ball rolls) will not. The ratio H/L is also equal to sin(θ) where θ is the angle the board makes with the ground.
A graph of acceleration vs. sin(θ) or acceleration vs. height/length will be linear. If the data collected from the experiment is good, the slope of the line should equal g.

Forces

The diagram below is a free body diagram of an object on an inclined plane:

There are many things to take note of in the diagram above as they relate to the motion of the object. First, the weight of the object (labeled mg) is directed downward as it always should. The normal force is perpendicular to the surface of the inclined plane. It is a situation such as this that the weight and normal force and not equal but opposite each other. For an inclined plane, the normal force will always be lesser than the weight, and in fact inspection of the diagram shows that the normal force is equal to mgcos(θ) where θ is the angle the inclined plane makes with the ground. The triangular shape of the inclined plane is a similar triangle to that made by the weight, and its components mgcos(θ) and mgsin(θ).

In simpler problems, the inclined plane may be described as frictionless, in which case the force of friction (labeled f) does not appear. In such a scenario, mgsin(θ) is wholly responsible for the obect's acceleration down the inclined plane. If friction is to be accounted for, then things change. First, the object can only accelerate if mgsin(θ) is greater than f. If mgsin(θ) equals f, then the object is either stationary or moving at a constant speed - either are non-acceleration scenarios. The value for the coefficient of friction can also be determined since μ equals friction divided by the normal force. If the object is not accelerating, substitution allows for the equivalence of μ to mgsin(θ) divided by mgcos(θ).

Simple Machine

The inclined plane is one of the classic examples of a simple machine - a tool that reduces the force needed to accomplish a certain quantity of work. Of course since work must be conserved, the reduction in force comes at the expense of applying the force over a greater distance. For example, the work needed to lift a 50 N box a vertical height of 1 meter requires 50 J of work. However, a board may be used to create an inclined plane. The longer the board, the less force will be needed, but the force will need to be applied the additional distance. Assume the length of the inclined plane is 5 m. Since 50 J of work must be accomplished to move the box, a force of 10 N must be applied (10 N times 5 m = 50 J). The (ideal) mechanical advantage for this incline plane is said to be 5. This is computed as the ratio of the length of the inclined plane to its vertical height (which still must be the original 1 m).

Experimental Investigation

A wealth of information regarding an inclined plane and its usefulness can be obtained by a rather simple exercise. Place an object of known mass on a board. Mark off the distance from the bottom edge of the object to the opposite end of the board. Slowly raise the end of the board at which the object rests, so an inclined plane is created. Continue to raise the board until the object begins to slide. Once the object starts sliding, measure the vertical height from the object's starting position on the board to the floor. These three mesurements - sliding distance of the object (the inclined plane's hypotenuse), the vertical height (the side opposite the angle of the inclined plane), and the object's mass can be used to analyze the concepts of friction, as well as the inclined plane as a simple machine.

Let's assume a 12 kg object begins to slide down an inclined plane of length 1.25 m when raised 37 cm off the ground. If one wants to calculate the frictional force acting against the sliding object, reconsider the free body doagram from earlier and shown again above. Note the frictional force (labeled f) is opposite the horizontal component of weight (mgsinθ). If the object is sliding at a constant velocity - not accelerating - then the frictional force can be determined. First, the angle of the inclined plane must be determined as the frictional force is equal to mgsinθ. Since the length of the slope and vertical height of the inclined plane are known, trigonometry can be utilized to find θ. Furthermore, since the slope and vertical height are the hypotenuse and opposite side of the right triangle created by the inclined plane, the sine function specifically is of use here:

Now the frictional force can be calculated. Substituting into mgsinθ, one will find the product of 12 kg, g (9.8 m/s2), and sinθ (0.292) for a value of 34.3 N.

From here the coefficient of friction μ can be calculated as well. Recall that the frictional force is also the product of the normal force and the coefficient of friction. Since the normal force is equal to mgcosθ and the frictional force has been solved for previously, ν can be found rather easily. A little algebra simplifies the problem further and shows that μ is in fact equal to tanθ:

This same information can be used to investigate the mechanical advantages of the inclined plane. First, the ideal mechanical advantage (IMA) is a ratio of the length of the slope to the vertical height of the plane. Essentially, the IMA gives the factor by which the force needed to accomplish a task should ideally be reduced. This comes at the expense at applying the reduced force over a longer distance. Hence, the formula and solution for IMA:

The actual mechanical advantage is a ratio of the work needed to push the object up the inclined plane to the work done by lifting the object a distance through the vertical height. If the inclined plane was 100% efficient this would equal the ratio of the weight of the object to the frictional force. This is never the case since the force to push the object up the incline will never be equal to the frictional force (although ideally it should).