Question

A 100-kg object rests on an inclined plane at an angle of ${30}^{\circ }$ to the floor. Find the components of the force perpendicular to and parallel to the plane. (The vertical component of the force exerted by an object of mass $m$ is its weight, which is $mg$, where $g=9.8\mathrm{m}/{\mathrm{s}}^2$ is the acceleration due to gravity.)

Short answer

Answer: The component of force perpendicular to the inclined plane is 848.6 N, and the component of force parallel to the inclined plane is 490 N.

Step-by-step solution

Calculate Weight Force (Fw)

Since we're given the object's mass (100 kg) and the acceleration due to gravity ($9.8\mathrm{m}/{\mathrm{s}}^2$), we can calculate the object's weight force (Fw) using the formula Fw = m * g, so: Fw = 100 kg * 9.8 $\mathrm{m}/{\mathrm{s}}^2$ = 980 Newtons (N)

Calculate Force Perpendicular to the Plane (Fn)

We can calculate the component of the force perpendicular to the inclined plane by using the cosine of the angle (30°) of the plane and the weight force: Fn = Fw * cos(30°) First, we find the cosine of the angle: cos(30°) = $\frac{\sqrt{3}}{2}$ Now, plug in the values for the weight force and the cosine of the angle: Fn = 980 N * $\frac{\sqrt{3}}{2}$ = 980 N * 0.866 Fn = 848.6 N So, the component of force perpendicular to the plane is 848.6 Newtons.

Calculate Force Parallel to the Plane (Fp)

Now, we can find the component of the force parallel to the inclined plane using the sine of the angle of the plane and the weight force: Fp = Fw * sin(30°) First, find the sine of the angle: sin(30°) = $\frac{1}{2}$ Now, plug in the values for the weight force and the sine of the angle: Fp = 980 N * $\frac{1}{2}$ = 980 N * 0.5 Fp = 490 N So, the component of force parallel to the plane is 490 Newtons. To summarize, the components of force are: - Perpendicular component (Fn): 848.6 N - Parallel component (Fp): 490 N

Key concepts

Force Components

When dealing with inclined planes in physics, breaking down the forces acting on an object is crucial. To understand how these forces work, it's important to know about force components. Imagine an object resting on a slope. The gravitational force acting on this object can be resolved into two components:

• Perpendicular to the plane
• Parallel to the plane

This process makes it easier to analyze how the object interacts with the inclined plane. By understanding these components, you can determine the force that tries to slide the object down the ramp and the force that pushes it into the ramp.
The perpendicular component (often labeled as Fn) is directed upwards and counteracts the force of gravity pressing the object into the slope. The parallel component (often labeled as Fp) acts down the slope, attempting to pull the object along the incline. Knowing these components helps in solving various physical problems associated with inclined planes.

Trigonometric Functions

Trigonometric functions play a vital role in physics, especially when dealing with angles and components of forces. For an inclined plane problem, the angle of the incline, usually measured from the horizontal, is crucial in determining these force components.
In our scenario, the sine and cosine functions are particularly useful. They help to resolve the weight of the object into components along and perpendicular to the incline. For a given angle $\theta$:

• The cosine function, $\cos (\theta )$, helps find the perpendicular component: $Fn=Fw\cdot \cos (\theta )$.
• The sine function, $\sin (\theta )$, is used to find the parallel component: $Fp=Fw\cdot \sin (\theta )$.

In essence, the use of trigonometric functions allows you to break down complex forces into simpler, more manageable parts. This translates a force in one direction into parts acting in two different directions, aligned with the inclined plane's geometry.

Newton's Laws of Motion

Understanding inclined plane problems also requires a grasp of Newton's Laws of Motion. These laws serve as the foundation of classical mechanics and explain how forces affect the motion of objects.
Newton's First Law, the law of inertia, tells us that an object at rest will stay at rest unless acted upon by a net external force. In the case of an object on an inclined plane, if there are no unbalanced forces, the object remains in place.
Newton's Second Law is often expressed as the equation $F=ma$ (force equals mass times acceleration). It explains how unbalanced forces cause changes in an object's motion. On an incline, this is essential when considering the forces leading the object to accelerate down the slope.
Finally, Newton's Third Law states that for every action, there is an equal and opposite reaction. This law is evident when the inclined plane pushes back against the object, coming from the normal force. By understanding these laws, you can predict and calculate the object's behavior through the forces acting upon it on the inclined plane.