A car is moving at the rate of 80km/h. the diameter of its wheel is 60cm. a, find the number of revolutions per minute that the wheels are rotating. b, find the angular speed of the wheels in radians per second.
60 cm = .6 m
one rotation = .6π m
80 km/h = 80,000 m/60 min = 400/3 m/min
number of rotations/min
= (400/3) / (.6π)
= 70.7 rotations in a minute
or 70.7/60 rotations per minute
= 1.1789 rotations per second
but 1 rotation = 2π radians,
so the angular velocity = 2π(1.1789..) radians/sec
= appr 7.4 radians/sec
check my arithmetic
To find the number of revolutions per minute that the wheels are rotating, we need to calculate the distance covered by the car in one revolution.
a) Distance covered by one revolution = circumference of the wheel
circumference = pi * diameter
circumference = pi * 60cm ≈ 188.5 cm
Next, we need to convert the speed of the car from km/h to cm/min (since we are finding the number of revolutions per minute):
Speed of the car = 80 km/h
Speed in cm/min = 80 km/h * 100,000 cm/km * 1 h/60 min
Speed in cm/min ≈ 133,333.33 cm/min
Finally, to find the number of revolutions per minute, we divide the speed of the car by the distance covered per revolution:
Number of revolutions per minute = Speed in cm/min / Distance covered by one revolution
Number of revolutions per minute = 133,333.33 cm/min / 188.5 cm ≈ 707.11 revolutions/min
Therefore, the wheels are rotating approximately 707.11 revolutions per minute.
b) To find the angular speed of the wheels in radians per second, we need to convert the speed from km/h to radians per second.
First, convert the speed from km/h to m/s:
Speed in m/s = 80 km/h * (1000 m/km) / (60 s/min) ≈ 22.22 m/s
The angular speed can be found using the formula:
Angular speed = Linear speed / radius of the wheel
The radius of the wheel is half the diameter:
Radius = 60cm / 2 = 30cm = 0.3m
Angular speed = 22.22 m/s / 0.3m ≈ 74.07 rad/s
Therefore, the angular speed of the wheels is approximately 74.07 radians per second.
To find the number of revolutions per minute, you need to determine the distance traveled by the car in one minute and then divide it by the circumference of the wheel.
a) First, calculate the distance traveled in one minute. Since the car is traveling at a rate of 80 km/h, in one minute it would cover 80/60 = 1.33 km (or 1330 meters).
b) Next, we need to find the circumference of the wheel. The diameter is given as 60 cm, so the radius is half of that, which is 30 cm or 0.3 meters. The circumference of a circle is given by the formula C = 2πr. Hence, the circumference of the wheel is 2π(0.3) = 1.88 meters.
c) Now, divide the distance traveled in one minute (1330 meters) by the circumference of the wheel (1.88 meters) to find the number of revolutions per minute. Therefore, 1330 / 1.88 = 708.5 revolutions per minute.
To find the angular speed of the wheels in radians per second, you need to divide the number of revolutions per minute by the conversion factor.
d) There are 60 seconds in a minute, so a minute has 2π radians (one full revolution). Therefore, the conversion factor is 2π/60 = π/30 (approximately 0.1047).
e) Finally, multiply the number of revolutions per minute (708.5) by the conversion factor (π/30) to find the angular speed in radians per second. Hence, the angular speed is approximately 708.5 * (π/30) = 74.4 radians per second.