Aparticle is moving on the perimeter of a circle with radius r=6 with angular speed of π/8 radians per second. after completing 5 full rotations, the particle traveled for 2 more seconds and stopped. what is the length of the total distance the particle traveled?

The total distance the particle traveled is 61.5π

Regular circular motion occurs when particles move in a circular path at a constant rate

The speed that is in the direction of the circle is called linear velocity

Can be formulated:

r = circle radius

T = period, the time required to take one round

or:

s = distance traveled by a particle, in circular motion = circumference of a circle = 2.π.r

t = travel time

In this circular motion the magnitude of angle taken is always the same.

The angle can be expressed in radians

2π radians = 360°

1 radians = 180 /π = 57.3°

The speed traveled by this angle is called the angular velocity (ω), which is the change in angle at each unit of time

Can be formulated:

ω = 2π/T

The relationship between linear velocity and angular velocity is:

v = ω.r

A particle moves for 5 rotations, meaning 5 times around the circle. So the distance traveled:

S1 = 5 x 2π. r

S1 = 5.2π.6

S1 = 60π

Linear speed of particles:

v = ω.r

v = π/8. 6

When moving 2 s, the distance traveled by the particle is:

S2 = v. t

S2 = 6/8π. 2

S2 = 12/8π = 3/2π

So the total path taken by the particle:

Total distance = S1 + S2

Total distance = 60π + 3/2π = 61.5π

the average velocity

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resultant velocity

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velocity position

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Keywords: regular circular motion, radians, circumference, distance traveled

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step-by-step explanation: