A stone whirled at the end of a rope 30 cm long, make 10 complete revolutions in 2 seconds. Find (i)the angular velocity in radians per seconds (ii)the linear speed (iii)the distance covered in 5 seconds.
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To find the answers to these questions, let's break down the problem step by step.
(i) The angular velocity is the rate at which an object rotates or spins. It is usually measured in radians per second.
To find the angular velocity, we use the formula:
Angular velocity (ω) = (2πn) / t,
where n is the number of revolutions and t is the time taken.
Given that the stone makes 10 complete revolutions in 2 seconds, we have:
n = 10 and t = 2.
Substituting these values into the formula, we get:
Angular velocity (ω) = (2π * 10) / 2.
Simplifying the equation further gives:
Angular velocity (ω) = 10π radians per second.
Therefore, the angular velocity in radians per second is 10π radians per second.
(ii) The linear speed is the speed at which an object moves along a circular path. It is usually measured in units such as meters per second (m/s).
To find the linear speed, we use the formula:
Linear speed (v) = ω * r,
where ω is the angular velocity and r is the radius of the circular path.
Given that the length of the rope is 30 cm, which is equivalent to 0.3 meters, we have:
r = 0.3 meters.
Substituting the angular velocity (ω = 10π radians per second) and the radius (r = 0.3 meters) into the formula, we get:
Linear speed (v) = (10π) * 0.3 meters per second.
Simplifying the equation further gives:
Linear speed (v) = 3π meters per second.
Therefore, the linear speed is 3π meters per second.
(iii) The distance covered in 5 seconds can be calculated using the formula:
Distance covered = Linear speed * time.
Given that the time is 5 seconds and the linear speed is 3π meters per second, we have:
Distance covered = (3π) * 5 meters.
Simplifying the equation further gives:
Distance covered = 15π meters.
Therefore, the distance covered in 5 seconds is 15π meters.