7 Facts On Relative Angular Velocity: With Problems


The angular velocity is the rate of rotations of the object in a rotational motion and the direction of angular velocity is always perpendicular to the plane of a rotating object.

The relative angular velocity is a ratio of the relative velocity between the two objects rotating along their axis of rotation divided by the total distance of separation between them. The relative angular velocity of the object varies with the distance of separation between the observer and the object.

What is relative angular velocity?

It is the angular velocity of one rotating object with respect to the motion of another object present in different co-ordinate of space and time.

The angular velocity of particle A with respect to another moving particle B is the rate of displacement of particle A with respect to particle B at that instant. The angular velocity is defined as the ratio of the component of the relative velocity of particle A with respect to particle B by the area of separation between both the particles.

Relative Angular Velocity Formula

The angular velocity is the ratio of the total displacement of an angle θ by the radii of an object or the distance between the centre and the circumference covered by the rotating object.

The formula to find the angular velocity is written as:

If the two objects are rotating with the angular velocity of

is separated at the distance ‘r’, then the relative angular velocity of the object with respect to another object is,

This formula gives the relative angular velocity between the two objects rotating at certain angular velocities separated at a distance ‘r’ away from each other.

How to find relative angular velocity?

The relative angular velocity of the object varies depending upon how far the object is located.

The relative angular velocity can be calculated by finding the relative speed of the object from the point of the reference and by measuring the distance of the object from the reference point using the formula ωR=VR*R, where VR is the relative angular velocity of the object measured in the reference frame.

If a particle A is moving along the positive x-direction with a velocity VA making an angle θ1 with the z-axis, and a particle B is moving along the positive y-direction with a velocity VB making an angle θ2 with the z-axis, then the relative angular velocity of the particle B with respect to particle A is,

Here, r is the distance between particle A and particle B.

The relative cosine velocities of the particles add up because both the particles are moving at a constant rate. The relative velocity between the two is the addition of their velocities.

Relative Angular Velocity Example

We know that the Earth is rotating as well as revolving around the Sun while the Moon is doing the same around the planet Earth. For us standing on the Earth, it would appear that the Earth is stationary while the Moon is revolving around the Earth.

relative angular velocity
Relative angular velocity of the Moon and the Earth; Image Credit: Pixabay

Hence, the relative speed between the Earth and the Moon with respect to our frame of reference is equal to the speed of the moon only. But for aliens standing on another planet and observing the Earth and Moon, they would say that the Earth and Moon both are moving with certain velocities relative to them.

The second example we can take is of ferries wheel. A person standing near the ferries wheel will observe that the wheel is rotating at a fast rate, while another person standing quite away from the wheel will notice that the angular velocity of a wheel is small as compared to the normal.

This is because the relative angular velocity of the ferries wheel with respect to the different frames of reference varies with distance.

Relative Angular Velocity Between Two Objects

Consider a child rotating her waist at a 360-degree rotational angle every second to keep the hula hoop of diameter 1 meter in momentum. The hula hoop rotates at an angle of 0.5π degrees per second. The direction of the angular velocity of both, the child and hula hoop is in the same direction. Then, the relative speed of the hula hoop and a child is,

Hence, the relative angular velocity of the hula hoop and a child is,

The relative angular velocity of the hula hoop is 270 rad/s.

The velocity of the propellers of a drone is dependent on the rotational speed of the motor it is connected to. The relative speed of a propeller with respect to the motor is actually zero because, for the motor, the propeller does not seem to be rotating.

For another propeller running in a counter-clockwise direction with the same rotational speed, the relative velocity of that propeller will be negative based on the direction.

Relative Angular Velocity of Geostationary Satellite

The geostationary satellites are mounted at a height of approximately 35.8k km above the ground surface. These satellites are specially meant to collect the data for a specific location and hence are planted by matching their revolving speed with the speed of the Earth.

For an observer on the Earth, the geostationary satellites appear to be stationary, as the speed of this satellite is synchronized with the speed of the Earth. Therefore, the relative velocity of the person standing on the Earth with respect to the geostationary satellite, and that of the geostationary satellite with respect to the person standing on the Earth is zero.

Hence, the relative angular momentum of the geostationary satellite with respect to the person standing on the Earth is also zero. But in the reference frame of the observer standing on the Moon, he will observe and calculate the speed of the geostationary satellite.

Relative Angular Velocity Direction

The direction of the angular velocity of the object is always along its axis of rotation.

If the object is rotating clockwise then the direction of an angular velocity is downward along the negative y-axis. And, if the rotational motion of the object is anti-clockwise, then the angular velocity of the object is upwards along the positive y-axis.

What is the direction of the relative angular velocity if the angular velocity of one object is 6iradand that of another object is [latex]-12\hat{i}\ rad/s[/latex]?

Given: The angular velocity of object 1 is, [latex]\omega _1=6\hat{i}\ rad/s[/latex]

The angular velocity of object 2 is, [latex]\omega _2=-12\hat{i}\ rad/s[/latex]

The negative sign indicates that the object is in a clockwise rotation and the direction of angular velocity is along the negative x-axis.

The relative angular velocity of both the spinning objects is,

[latex]\omega _R=\omega _1+\omega _2[/latex]

Substituting the values in this equation, we get:

[latex]\omega _R=\left ( 6\hat{i} -12\hat{i} \right )rad/s[/latex]

[latex]\omega _R=-6\hat{i}\ rad/s[/latex]

The relative angular velocity of both the objects is [latex]-6\hat{i}\ rad/s[/latex] and is along the negative x-axis.

What is the relative angular velocity of a boomerang moving at an angle 600 with a velocity of 3m/s with respect to the person standing stationary at a distance of 10 m?

Given: The velocity of a person is, [latex]v_1=0[/latex]

The velocity of a boomerang is, [latex]v_2=3\ m/s[/latex]

The angle of displacement of boomerang is, [latex]\theta =60^0[/latex]

Distance of separation, r=10m

The component of velocity of the boomerang is,

[latex]V=v_2cos60^0[/latex]

[latex]V=3\times \frac{1}{2}=1.5 m/s[/latex]

The expression to find the relative angular velocity of the boomerang with respect to a person stating stationary is,

[latex]\omega _R=\frac{V+v_1}{r}[/latex]

Substituting the values in this equation, we get:

[latex]\omega _R=\frac{1.5+0}{10}=0.15/sec[/latex]

Hence, the relative angular velocity of a boomerang for a person standing 10m away is 0.15/sec.

Conclusion

The relative angular velocity of the object in the frame S from the frame S’ is the ratio of the relative speed of the object measured by the observer in frame S’ divided by the distance of separation between the frame S and S’. The relative angular velocity of the object decreases as the distance of separation between both the frames increases.

AKSHITA MAPARI

Hi, I’m Akshita Mapari. I have done M.Sc. in Physics. I have worked on projects like Numerical modeling of winds and waves during cyclone, Physics of toys and mechanized thrill machines in amusement park based on Classical Mechanics. I have pursued a course on Arduino and have accomplished some mini projects on Arduino UNO. I always like to explore new zones in the field of science. I personally believe that learning is more enthusiastic when learnt with creativity. Apart from this, I like to read, travel, strumming on guitar, identifying rocks and strata, photography and playing chess. Connect me on LinkedIn - linkedin.com/in/akshita-mapari-b38a68122

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