Dynamics is one of the important branches of physics, which studies the causes of the motion of bodies in space. In this article, we consider from the point of view of theory one of the typical problems of dynamics - the motion of a body along an inclined plane, and also give examples of solutions to some practical problems.
The basic formula of dynamics
Before proceeding to the study of the physics of body motion along an inclined plane, we present the necessary theoretical information to solve this problem.
In XVII, Isaac Newton, thanks to practical observations of the movement of macroscopic surrounding bodies, deduced three laws that now bear his last name. All classical mechanics is based on these laws. We are interested in this article only the second law. Its mathematical form is given below:
F¯ = m × a¯.
The formula says that the action of an external force F¯ will give acceleration a¯ to a body of mass m. We will use this simple expression to solve the problems of motion of a body along an inclined plane.
Note that force and acceleration are vector quantities directed in the same direction. In addition, strength is an additive characteristic, that is, in the above formula, F¯ can be considered as the resulting effect on the body.
Inclined plane and forces acting on the body located on it
The key point on which the success of solving the problems of motion of a body along an inclined plane depends is the determination of the forces acting on the body. By the definition of forces is understood the knowledge of their modules and directions of action.
The figure below shows where the body (car) is at rest on a plane inclined at an angle to the horizon. What forces act on it?
The list below lists these forces:
- severity
- support reactions;
- friction;
- thread tension (if present).
Next, we describe in more detail each of them in relation to the problem under consideration.
The force of gravity
First of all, it is gravity (F g ). It is directed vertically down. Since the body has the ability to move only along the surface of the plane, when solving problems, the force of gravity is decomposed into two mutually perpendicular components. One of the components is directed along the plane, the other is perpendicular to it. Only the first of them leads to the appearance of acceleration in the body and, in fact, is the only driving factor for the body in question. The second component determines the occurrence of the reaction force of the support.
Prop reaction
The second force acting on the body is the support reaction (N). The reason for its appearance is associated with Newton’s third law. The value of N indicates the force with which the plane acts on the body. It is directed upward perpendicular to the inclined plane. If the body was on a horizontal surface, then N would equal its weight. In the case under consideration, N is only the second component obtained by decomposition of gravity (see the paragraph above).
The reaction of the support does not directly affect the nature of the movement of the body, since it is perpendicular to the plane of inclination. Nevertheless, it causes the appearance of friction between the body and the surface of the plane.
Friction force
The third force that should be considered when studying body motion on an inclined plane is friction (F f ). The physical nature of friction is not simple. Its appearance is associated with microscopic interactions of contacting bodies having inhomogeneous contact surfaces. There are three types of this force:
The friction of rest and sliding are described by the same formula:
F f = µ × N,
where µ is a dimensionless coefficient, the value of which is determined by the materials of rubbing bodies. So, with friction of a tree sliding on a tree, µ = 0.4, and ice on ice - 0.03. The coefficient for rest friction is always greater than that for sliding.
The rolling friction is described by a different formula from the previous one. It has the form:
F f = f × N / r.
Here r is the radius of the wheel, f is the coefficient having the dimension of the inverse length. This friction force, as a rule, is much less than the previous ones. Note that its value is affected by the radius of the wheel.
The force F f , whatever type it is, is always directed against the movement of the body, that is, F f seeks to stop the body.
Thread tension
When solving the problems of body motion on an inclined plane, this force is not always present. Its appearance is determined by the fact that a body located on an inclined plane is connected with an inextensible thread to another body. Often the second body hangs on a thread through a block outside the plane.
On the object located on the plane, the force of the tension of the thread acts either accelerating it or slowing it down. It all depends on the modules of forces acting in the physical system.
The appearance of this force in the problem significantly complicates the solution process, since it is necessary to consider simultaneously the motion of two bodies (on the plane and hanging).
Next, we give an example of solving two problems without the participation of the thread tension force.
The task of determining the critical angle
Now it's time to apply the described theory to solve real problems of motion along the inclined plane of the body.
Suppose a timber bar has a mass of 2 kg. It is on a wooden plane. It should be determined at what critical angle of inclination of the plane the beam will begin to slide along it.
The glide of the beam will occur only when the total force acting down along the plane on it is greater than zero. Thus, to solve this problem, it is enough to determine the resulting force and find the angle at which it becomes greater than zero. According to the condition of the problem, only two forces will act on the beam along the plane:
- component of gravity F g1 ;
- rest friction F f .
To begin the gliding of the body, the condition must be met:
F g1 ≥ F f .
Note that if the component of gravity exceeds the static friction, then it will also be greater than the sliding friction force, that is, the movement that begins will continue with constant acceleration.
The figure below shows the directions of all acting forces.
We denote the critical angle by θ. It is easy to show that the forces F g1 and F f will be equal to:
F g1 = m × g × sin (θ);
F f = μ × m × g × cos (θ).
Here m × g is the body weight, µ is the coefficient of rest friction for a pair of wood-tree materials. From the corresponding coefficient table, you can find that it is equal to 0.7.
Substitute the found values into the inequality, we obtain:
m × g × sin (θ) ≥ µ × m × g × cos (θ).
Transforming this equality, we arrive at the condition for the motion of the body:
tg (θ) ≥ µ =>
θ ≥ arctan (μ).
We got a very interesting result. It turns out that the value of the critical angle θ does not depend on the mass of the body on an inclined plane, but is uniquely determined by the coefficient of rest friction µ. Substituting its value into the inequality, we obtain the value of the critical angle:
θ ≥ arctg (0.7) ≈ 35 o .
The task of determining the acceleration when moving along an inclined plane of the body
Now we solve a slightly different problem. Let a wooden beam be located on a glass inclined plane. The plane to the horizon is inclined at an angle of 45 o . It should be determined with what acceleration the body will move if its mass is 1 kg.
We write the main equation of dynamics for this case. Since the force F g1 will be directed along the movement, and F f against it, the equation will take the form:
F g1 - F f = m × a.
We substitute the formulas obtained in the previous problem for the forces F g1 and F f , we have:
m × g × sin (θ) - µ × m × g × cos (θ) = m × a.
Where do we get the formula for acceleration:
a = g × (sin (θ) - µ × cos (θ)).
Again we got a formula in which there is no body weight. This fact means that bars of any mass will slide off at the same time along an inclined plane.
Given that the coefficient µ for rubbing wood-glass materials is 0.2, we substitute all the parameters into equality, we get the answer:
a ≈ 5.55 m / s 2 .
Thus, the technique for solving problems with an inclined plane is to determine the resulting force acting on the body, and in the subsequent application of Newton’s second law.