a) The angular velocity is 0.004 rad/s
b) The linear velocity is [tex]6\cdot 10^{-4} m/s[/tex]
c) The period of motion is 235 s
Explanation:
a)
The angular velocity of an object in circular motion is the rate of change of the angular position of the object. It can be calculated as follows:
[tex]\omega=\frac{s}{t}[/tex]
where
s is the arc of length covered by the object
t is the time elapsed
For the stone moving in the horizontal circle in the problem, we have
s = 4 cm = 0.04 m is the length of the arc covered
t = 10 s is the time elapsed
Substituting, we find the angular velocity:
[tex]\omega=\frac{0.04}{10}=0.004 rad/s[/tex]
b)
In a circular motion, the linear velocity is related to the angular velocity by the equation
[tex]v=\omega r[/tex]
where
[tex]v[/tex] is the linear velocity
[tex]\omega[/tex] is the angular velocity
r is the radius of the circle
For the stone in this problem, we have:
[tex]\omega=0.004 rad/s[/tex] (angular velocity, calculated in part a)
r = 15 cm = 0.15 m (radius of the circle)
Substituting, we find
[tex]v=(0.004)(0.15)=6\cdot 10^{-4} m/s[/tex]
c)
In a circular motion, the period is the time taken for the object to complete exactly one revolution.
The circle in this problem has a radius of
r = 0.15 m
So, the circumference is
[tex]L=2\pi r = 2\pi(0.15)=0.94 m[/tex]
We know that the stone covers an arc length of s = 0.04 m in 10 s, therefore we can use the following proportion to find the time taken for the stone to cover one circumference (and this time is the period of the motion):
[tex]0.04 m: 10 s = 0.94 m : T\\T=\frac{(10s)(0.94m)}{0.04m}=235 s[/tex]
Learn more about circular motion:
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