Kinematics is a part of physics that considers the laws of motion of bodies. Its difference from the dynamics is that it does not consider the forces acting on the moving body. This article is devoted to the kinematics of rotational motion.
Rotational motion and its difference from translational
If you pay attention to the surrounding moving objects, you will notice that they either move in a straight line (the car goes along the road, the plane flies through the sky), or around the circumference (the same car that enters into a turn, wheel rotation). More complex types of movement of objects can be reduced to a first approximation to a combination of the two types noted.
Translational movement involves changing the spatial coordinates of the body. Moreover, it is often regarded as a material point (geometric dimensions are not taken into account).
Rotational motion is a type of movement in which the system moves in a circle around a certain axis. Moreover, the object in this case is rarely considered as a material point, most often they use another approximation - an absolutely solid body. The latter means that the elastic forces acting between the atoms of the body are neglected and it is believed that the geometric dimensions of the system do not change during rotation. The simplest case is a fixed axis.
The kinematics of translational and rotational motion obeys the same Newtonian laws. Similar physical quantities are used to describe both types of displacement.
What quantities describe motion in physics?
The kinematics of rotational and translational motion uses three basic quantities:
- Distance traveled. We will denote it by the letter L for translational and θ for rotational motion.
- Speed. For the linear case, it is usually prescribed with the Latin letter v, for movement along a circular path - with the Greek letter ω.
- Acceleration. For a linear and circular path, the symbols a and α are used, respectively.
The concept of a trajectory is also often used. But for the types of movement of objects under consideration, this concept becomes trivial, since translational motion is characterized by a linear trajectory, and rotational - by a circle.
Linear and angular speeds
We begin to consider the kinematics of the rotational motion of a material point with the concept of speed. It is known that for the translational movement of bodies, this quantity describes which path will be overcome in a unit of time, that is:
v = L / t
The value of v is measured in meters per second. For rotation, it is inconvenient to consider this linear velocity, since it depends on the distance to the axis of rotation. A slightly different characteristic is introduced:
ω = θ / t
This is one of the main formulas for the kinematics of rotational motion. It shows at what angle θ the whole system rotates around a fixed axis in time t.
Both of the above formulas reflect the same physical process of speed of movement. Only for the linear case is distance important, and for the circular case, the angle of rotation.
Both formulas interact with each other. Get this connection. If θ is expressed in radians, then a material point rotating at a distance R from the axis, having made one revolution, will go the path L = 2 * pi * R. The expression for linear velocity will take the form:
v = L / t = 2 * pi * R / t
But the ratio 2 * pi radian to time t is nothing but the angular velocity. Then we get:
v = ω * R
This shows that the greater the linear velocity v and the smaller the radius of rotation R, the greater the angular velocity ω.
Linear and angular acceleration
Another important characteristic in the kinematics of the rotational motion of a material point is angular acceleration. Before getting to know him, we give a formula for a similar linear quantity:
1) a = dv / dt
2) a = Δv / Δt
The first expression reflects instantaneous acceleration (dt -> 0), while the second formula is appropriate to apply if the speed changes uniformly over time Δt. The acceleration obtained in the second embodiment is called average.
Given the similarity of the quantities that describe the linear and rotational motion, for angular acceleration, we can write:
1) α = dω / dt
2) α = Δω / Δt
The interpretation of these formulas is exactly the same as for the linear case. The only difference is that a shows how many meters per second the speed per unit time changes, and α shows how many radians per second the angular velocity changes over the same time period.
Find the connection between these accelerations. Substituting the value for v expressed in terms of ω into any of the two equalities for α, we obtain:
α = Δω / Δt = Δv / Δt * 1 / R = a / R
It follows that the smaller the radius of rotation and the greater the linear acceleration, the greater the value of α.
Distance traveled and turning angle
It remains to give formulas for the last of the three basic quantities in the kinematics of rotational motion around a fixed axis — for the angle of rotation. As in the previous paragraphs, we first write down the formula for uniformly accelerated rectilinear movement, we have:
L = v 0 * t + a * t 2/2
A complete analogy with rotational motion leads to the following formula for it:
θ = ω 0 * t + α * t 2/2
The last expression allows you to get the angle of rotation for any time t. Note that the circle is 2 * pi radians (≈ 6.3 radians). If, as a result of solving the problem, the value of θ is greater than the indicated value, then the body has made more than one revolution around the axis.
We obtain the coupling formula L and θ by substituting the corresponding values for ω 0 and α through linear characteristics:
θ = v 0 * t / R + a * t 2 / (2 * R) = L / R
The resulting expression reflects the meaning of the angle θ itself in radians. If θ = 1 rad, then L = R, that is, an angle of one radian rests on an arc with a length of one radius.
Problem solving example
We solve the following problem of the kinematics of rotational motion: it is known that the car moves at a speed of 70 km / h. Knowing that the diameter of his wheel is D = 0.4 meters, it is necessary to determine the value of ω for him, as well as the number of revolutions that it will make when the car travels 1 kilometer.
To find the angular velocity, it is enough to substitute the known data in the formula for relating it to the linear velocity, we obtain:
ω = v / R = 7 * 10 4/3600 / 0.2 = 97.222 rad / s.
Similarly, for the angle θ, on which the wheel will turn, after passing 1 km, we get:
θ = L / R = 1000 / 0.2 = 5000 rad.
Given that one revolution is 6.2832 radians, we get the number of revolutions of the wheel, which corresponds to this angle:
n = θ / 6.2832 = 5000 / 6.2832 = 795.77 revolutions.
We answered questions using the formulas given in the article. It was also possible to solve the problem in a different way: calculate the time in which the car will travel 1 km, and substitute it in the formula for the angle of rotation, from where to obtain the angular velocity ω. The answer is found.