In circular motion, What does it mean s = r.θ?

In summary, the equation s = r.θ represents the relationship between arc length and angular displacement in circular motion, where s is the arc length, r is the radius, and ø is the angle measure in radians. This equation can be derived by assuming that the arc length is directly proportional to the angle and using the known values for a full rotation (2π radians) to determine the constant of proportionality as the radius of the circle.
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In the circular motion, What does it mean by this equation s = r.θ?
 
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S=rø is the formula for arc length, or angular displacement (ø rad).

R is the radius
Ø is the angle measure in radians.
 
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arctheta.png


The scheme is not perfect but I think you get the idea...

If you want to know how this equation is proved:

Due to symmetry of the circle we can assume that the arc length ##s## will be directly proportional to the angle ##\theta##. So it will be

##s=k\theta## (1) for some constant k.

But we know for angle ##2\pi## corresponds arc length ##2\pi r##. So if we apply equation (1) for those pair of data we get

##2\pi r=k (2\pi)## from which we can deduce that the constant k is actually the radius of the circle, ##k=r##.
 

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1. What is the meaning of "s = r.θ" in circular motion?

In circular motion, "s = r.θ" represents the relationship between the arc length (s), radius (r), and central angle (θ) of a circle. It is a mathematical expression known as the arc length formula, which is used to calculate the distance traveled by an object in circular motion.

2. How is the arc length formula derived?

The arc length formula is derived from the basic geometry of a circle. The circumference of a circle is equal to 2πr (where r is the radius), and the central angle θ is a fraction of the total angle of a circle (360 degrees). Therefore, we can multiply the circumference by the fraction θ/360 to get the arc length formula: s = (2πr) x (θ/360).

3. Can the arc length formula be used for any type of circular motion?

Yes, the arc length formula can be used for any type of circular motion, as long as the motion follows a circular path. This includes uniform circular motion (where the object moves at a constant speed) and non-uniform circular motion (where the speed of the object changes).

4. What are the units of measurement for each variable in the arc length formula?

The units of measurement for the arc length formula depend on the units used for the variables. The arc length (s) is typically measured in meters (m), the radius (r) is measured in meters (m), and the central angle (θ) is measured in degrees (°) or radians (rad). It is important to ensure that all units are consistent when using the arc length formula.

5. How is the arc length formula used in real-world applications?

The arc length formula is used in many real-world applications, such as calculating the distance traveled by a car on a curved road, the distance traveled by a satellite in orbit, and the distance traveled by a Ferris wheel. It is also used in engineering and design to determine the length of an arc needed for a specific curve or to calculate the amount of material needed for a curved structure.

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