A person rotates in circular path having radius 5 m calculate the average speed velocity in 5 second for complete 1 rotation also calculate the displacement covered by a person
Answers
Explanation:
Rotation Angle
When objects rotate about some axis—for example, when the CD (compact disc) in Figure 1 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle Δθ to be the ratio of the arc length to the radius of curvature:
Δ
θ
=
Δ
s
r
The figure shows the back side of a compact disc. There is a scratched part on the upper right side of the C D, about one-fifth size of the whole area, with inner circular dots clearly visible. Two line segments are drawn enclosing the scratched area from the border of the C D to the middle plastic portion. A curved arrow is drawn between the two line segments near this middle portion and angle delta theta written alongside it.
Figure 1. All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle Δθ in a time Δt.
A circle of radius r and center O is shown. A radius O-A of the circle is rotated through angle delta theta about the center O to terminate as radius O-B. The arc length A-B is marked as delta s.
Figure 2. The radius of a circle is rotated through an angle Δθ. The arc length Δs is described on the circumference.
The arc length Δs is the distance traveled along a circular path as shown in Figure 2 Note that r is the radius of curvature of the circular path.
We know that for one complete revolution, the arc length is the circumference of a circle of radius r. The circumference of a circle is 2πr. Thus for one complete revolution the rotation angle is
Δ
θ
=
2
π
r
r
=
2
π
.
This result is the basis for defining the units used to measure rotation angles, Δθ to be radians (rad), defined so that 2π rad = 1 revolution.
A comparison of some useful angles expressed in both degrees and radians is shown in Table 1.
Table 1. Comparison of Angular Units
Degree Measures Radian Measure
30º
π
6
60º
π
3
90º
π
2
120º
2
π
3
135º
3
π
4
180º π
A circle is shown. Two radii of the circle, inclined at an acute angle delta theta, are shown. On one of the radii, two points, one and two are marked. The point one is inside the circle through which an arc between the two radii is shown. The point two is on the cirumfenrence of the circle. The two arc lengths are delta s one and delta s two respectively for the two points.
Figure 3. Points 1 and 2 rotate through the same angle (Δθ), but point 2 moves through a greater arc length (Δs) because it is at a greater distance from the center of rotation (r).
If Δθ = 2π rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are 360º in a circle or one revolution, the relationship between radians and degrees is thus 2π rad = 360º so that
1
rad
=
360
∘
2
π
≈
57.3
∘
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