The branch of physics that studies the motion of bodies along various paths is called kinematics. Practically useful types of moving objects are moving in a straight line and in a circle. We consider in the article what these types of motion are, what formulas they are described, and also give a relationship of angular and linear velocity.
Straight ahead
The relationship of angular and linear velocity can be determined if you know what quantities are involved. Let's start with the linear speed.
From school, everyone knows that the movement of objects in space is characterized by three main quantities:
- distance traveled S;
- travel time t;
- speed v.
The formula linking these quantities into a single equality is given below:
S = v * t.
The above expression describes the uniform motion of the body in a straight line. In the international SI system of units, S is measured in meters (m), t - in seconds (s), v - in meters per second (m / s). In addition to these units, the path and time can be measured in kilometers (km) and hours (h), respectively. Then the speed will be expressed in kilometers per hour (km / h).
The written formula can be used to solve a wide range of practical problems, for example, the movement of vehicles on roads, the movement of ships and boats on rivers, the flight of birds and so on.
Circular motion
Before proceeding to the derivation of the relation between linear and angular velocity, the latter should be considered from the point of view of physics.
Angular velocity appears in physics when it comes to rotating objects. Examples are the rotation of a bicycle wheel, a flywheel of a car, or a planet around its star. The angular velocity of the body shows at what angle in radians it rotates per unit of time. Usually this value is denoted by the Greek letter ω (omega). It is measured in radians per second (rad / s).
By analogy with the linear case, there are three main quantities that describe circular motion with a constant angular velocity:
- rotation angle θ;
- time t;
- angular velocity ω.
The corresponding formula that relates these quantities is as follows:
θ = ω * t.
The angle of rotation of the body θ about the axis of rotation is measured in radians. Recall that the circle has 2 * pi radians (about 6.28). If the θ value obtained by the formula turned out to be more than 2 * pi, then this means that the body made more than one revolution around the axis.
Thus, the written expression allows you to calculate the number of revolutions performed by the body for a known period of time t.
Correlation of angular and linear velocity
Now you can consider this issue. Suppose that a body with linear velocity v rotates in a circle of radius R. To get a connection between linear and angular velocity, we consider how much time it will take for the body to make a complete one revolution. Since the distance traveled will be equal to the circumference, the following expression will be true:
t = S / v = 2 * pi * R / v.
Now we use the angular values. For the found time of one revolution t, the body will turn exactly 2 * pi radians. The latter means that its angular velocity will be equal to:
ω = θ / t = 2 * pi / t.
We substitute the time t calculated above and obtain the relation between the angular and linear velocities:
ω = 2 * pi / t = 2 * pi / (2 * pi * R / v) = v / R.
The resulting formula can be written in two forms:
ω = v / R;
v = ω * R.
Each of the expressions is applied depending on what value in the problem statement is known. The formulas allow us to make an important conclusion: the larger the rotation radius of the orbit, the greater the linear velocity at a constant angular velocity.
Next, we solve an interesting problem on the application of the obtained formulas.
Which is faster - Earth or Mars?
It is known that the Earth and Mars are the 3rd and 4th planets of the solar system, respectively. Both planets move in approximately circular orbits. The distance from our star to the Earth is 149 597 870.691 km, and it makes one revolution around it in 365.256 days. Mars is located at a distance of 227,936,640 km from the Sun, and makes one revolution around it in 686,971 Earth days. It is necessary to determine and compare the linear speeds of the planets.
The angular velocity of the planet can be calculated by the formula:
ω = 2 * pi / T.
Where T is the period (time of one revolution around the star). Substituting ω into the formula for v, we obtain:
v = 2 * pi * R / T.
We translate the time of planetary rotation into hours and substitute the data in this equality, we get:
- for the Earth: v = 2 * 3.14 * 149597870.691 / (365.256 * 24) ≈ 107.2 thousand km / h;
- for Mars: v = 2 * 3.14 * 227936640 / (686.971 * 24) ≈ 86.8 thousand km / h.
Both numbers are huge. So, the Earth flies in space in one hour, a distance almost equal to its three circles at the equator. The speeds obtained indicate that the Earth moves faster than Mars, and its speed is 24% more than Martian.