Question

A particle 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?

Answer 1

First of all, let's calculate the the circumference. This is found by this equation:

$P=2\pi r \\ \\ Since \ r=6 \\ \\ P=2\pi \times 6 \\ \\ \therefore P=12\pi$

The distance of completing 5 full rotations is:

$D_{5}=5P \\ \\ \therefore D_{5}=5(12\pi) \\ \\ \therefore D_{5}=60\pi \ radians$

In one second the particle traveled:

$\frac{\pi}{8} \ radians$

Since the particle traveled for 2 more seconds and stopped, this distance is:

$2\left(\frac{\pi}{8}\right) \\ \\ \frac{\pi}{4} \ radians$

Finally, the length of the total distance the particle traveled is:

$D=60\pi+\frac{\pi}{4} \\ \\ \therefore \boxed{D=\frac{241}{4}\pi}$

Answer 2

The total distance the particle traveled is 61.5π

Further explanation

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:

$\large{\boxed{\bold{v=\frac{2\pi\:r }{T}}}}$

r = circle radius

T = period, the time required to take one round

or:

$v=\frac{s}{t}$

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

$v=\frac{6}{8}\pi$

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

Learn more

the average velocity

brainly.com/question/5248528

resultant velocity

brainly.com/question/4945130

velocity position

brainly.com/question/2005478

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