Angular Position – Theta

Physics> Angular Position – Theta

Angle of rotation is a measurement of the amount (angle) at which a figure rotates around a fixed point (often in the central part of a circle).

Learning challenge

• Trace the relationship between radians on a CD.

Key points

• The arc length Δs is the distance traveled along a circular path. R is the radius of curvature of the circular path.
• Angle of rotation is the amount of rotation that corresponds to the linear distance. Let us calculate the angle of rotation Δθ as the ratio of the arc length to the radius of curvature: Δθ = Δs / r.
• For one complete revolution, the angle of rotation is 2π.

Term

• Angular position – the angle in radians (degrees, revolutions) through which the point or line is rotated in a certain direction around the specified axis.

When objects rotate around an axis (CD rotation), each point follows an arc of a circle. Let’s trace a line from the center of the disc to the edge. Each groove used to record sound along a line moves through one angle and time interval.

Angle of rotation is the amount of rotation that corresponds to the linear distance. Let us calculate the angle of rotation Δθ as the ratio of the arc length to the radius of curvature:

Δθ = Δs / r

All points on CDs move in circular arcs. Depressions along the lines from the center to the edge move through one angle Δ for the period Δt

In mathematics, angular position is a measurement of the amount at which an object makes a revolution around a fixed point.

The radius of the circle is rotated by an angle Δ. The arc length Δs is described along a circle

The arc length Δs is the distance traveled along a circular path. R is the radius of curvature of the circular path. For a full turnover, the arc length is the volume of a circle of radius r. Circle volume = 2πr. It turns out that for a full turnover the angle of rotation is:

Δθ = (2πr) / r = 2π.

This is the basis for calculating the units used when measuring the angular position of radians:

2π rad = 1 revolution.

If Δθ = 2π rad, then the CD has completed a full rotation, and each point on the CD has returned to its original position. Since one circle is 360 °, the ratio between radians and degrees is 2π rad = 360 °, so:

1rad = 360 ° / 2π = 57.3 °.

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