2 racing cars of masses m1 and m2 are moving in circles of radii r1 and r2 respectively. Their speeds are such that each makes a complete circle in same time t. The ratio of Angular speed of the first car to the second car is?
To find the ratio of the angular speeds of the two racing cars, we should first understand the relationship between angular speed, radius, and linear speed.
The angular speed (ω) is defined as the rate at which an object rotates or revolves around an axis. It is given by the formula:
ω = v / r
where v is the linear speed and r is the radius of the circular path.
In this scenario, each car makes a complete circle in the same time, t. This indicates that the time taken for one complete revolution is the same for both cars.
Let's assume the linear speed of the first car is v1 and the linear speed of the second car is v2. Given that both cars complete one revolution in the same time, the time taken for one revolution (t) is constant for both cars.
Since the time (t) is constant, we can equate the distances traveled by each car in one revolution.
The distance traveled by the first car in one revolution is equal to the circumference of its circular path: 2πr1.
Similarly, the distance traveled by the second car in one revolution is equal to the circumference of its circular path: 2πr2.
Equating the distances traveled, we have:
2πr1 = 2πr2
Dividing both sides of the equation by 2π, we get:
r1 = r2
This implies that the radii of the circular paths for both cars are equal.
Now, using the relationship between linear speed, angular speed, and radius mentioned earlier, we can write the equations:
ω1 = v1 / r1
ω2 = v2 / r2
Since r1 = r2, we can simplify the equations:
ω1 = v1 / r
ω2 = v2 / r
Dividing ω1 by ω2, we get:
ω1 / ω2 = (v1 / r) / (v2 / r)
The radii (r) cancel out, and we are left with:
ω1 / ω2 = v1 / v2
Therefore, the ratio of the angular speed of the first car (ω1) to the second car (ω2) is equal to the ratio of their linear speeds (v1 / v2).