To find the answer to this question, we need to understand the relationship between speed, distance, and time for a rotating object. The key concept here is the formula for calculating the circumference of a circle, which is C = πd, where C is the circumference and d is the diameter.
(a) To find out how fast the wheel is turning, we need to calculate its angular speed, which is usually represented by the Greek letter ω (omega). Angular speed is defined as the rate at which an object rotates or turns.
The formula for angular speed is given by ω = v / r, where ω is the angular speed, v is the linear velocity (speed), and r is the radius of the wheel.
First, let's calculate the radius of the wheel. The radius (r) is equal to half the diameter (d), so r = d/2.
Given that the diameter (d) is 50 cm, the radius is 50 cm / 2 = 25 cm = 0.25 m.
We are also given that the speed (v) of the wheel is 20 m/sec.
Using the formula ω = v / r, we can substitute the values to find the angular speed:
ω = 20 m/sec / 0.25 m = 80 radians/sec
Therefore, the wheel is turning at a speed of 80 radians/sec.
(b) To find the number of revolutions the wheel will make, we need to first calculate the distance traveled by the wheel as it comes to rest. Since the wheel is coasting uniformly, it will undergo constant deceleration.
The formula for distance covered during uniformly decelerated motion is given by s = ut + (1/2)at^2, where s is the distance covered, u is the initial velocity, t is the time of travel, and a is the deceleration.
Given that the initial velocity (u) is 20 m/sec, the time of travel (t) is 30 sec, and the wheel is coming to rest, meaning the final velocity (v) is 0 m/sec, we can rewrite the formula as:
s = ut + (1/2)a t^2
Since the wheel is coming to rest, the final velocity (v) is 0 m/sec, so the change in velocity (Δv) can be calculated as:
Δv = v - u
Substituting the values:
Δv = 0 - 20 = -20 m/sec
Using the formula for deceleration (a = Δv / t), we can find the deceleration:
a = -20 m/sec / 30 sec ≈ -0.67 m/sec^2 (negative sign indicates deceleration)
Now we have all the values we need to find the distance covered (s). Substituting the values into the formula:
s = ut + (1/2)a t^2
s = 20 m/sec * 30 sec + (1/2)(-0.67 m/sec^2)(30 sec)^2
Simplifying further:
s = 600 m - 300 m ≈ 300 m
Since the circumference of the wheel (C) is equal to the distance covered (s) when the wheel makes one full revolution, we can calculate the number of revolutions (n) as:
n = s / C
Given that the distance covered is approximately 300 m and the diameter of the wheel is 50 cm, we can find the circumference (C):
C = πd = π * 50 cm = 157.08 cm ≈ 1.57 m
Now we can substitute the values into the formula:
n = 300 m / 1.57 m ≈ 191.08
Therefore, the wheel will make approximately 191 revolutions before coming to rest.