a bicycle with 24inch diameter wheels is traveling at 15mi/hr.
a) Find the angular speed of the wheels in rad/min.
b)How many revolutions per minute do the wheels make?
Diameter = 24 in
Radius = 12 in
(a) angular speed (ω) = ?
We use the formula:
v = ω r
Substituting,
15 mi/hr = ω * 12 in
Note that 1 mile = 63360 inches, and 1 hour = 60 min:
15 mi * (63360 in / mi) / (1 hr * 60 min / hr) = ω * 12 in
15840 in / min = ω * 12 in
ω = 15840 / 12
ω = 1320 rad/min
(b) rev/min = ?
Note that 1 revolution = 2π. Thus,
1320 rad/min * 1 rev / 2π rad = 210.08 rev/min
hope this helps~ `u`
Thank you Jai!
a) Well, let me put on my math clown nose for a moment! To find the angular speed in rad/min, we need to convert the linear speed to angular speed. The formula for angular speed is:
Angular speed = Linear speed / Radius
The radius of the wheel is half its diameter, which is 12 inches or 1 foot. So, the linear speed is 15 mi/hr or (15 * 5280) ft/hr. Converting this to ft/min, we get (15 * 5280) / 60 ft/min. So, the angular speed is:
Angular speed = (15 * 5280) / 60 / 1 ft/min = 880 ft/min
To convert this to rad/min, we need to remember that 1 revolution is equal to 2π radians. So, the angular speed is:
Angular speed = 880 ft/min / (2π * 1 ft) = 440π rad/min
b) Now, let's clown around with revolutions per minute! To find the number of revolutions per minute, we need to divide the linear speed by the circumference of the wheel. The formula for revolutions per minute is:
Revolutions per minute = Linear speed / Circumference
The circumference of a wheel is equal to its diameter times π (π is approximately 3.14159). So, the circumference is 24 inches * π, which is approximately 75.398 inches. Converting this to feet, we get 75.398 inches / 12 inches/ft = 6.283 ft. So, the number of revolutions per minute is:
Revolutions per minute = 15 mi/hr / 6.283 ft/rev = 2.387 rev/min
So, the angular speed of the wheels is approximately 440π rad/min, and the wheels make approximately 2.387 revolutions per minute. Keep on pedaling!
To find the angular speed of the wheels in rad/min, we need to convert the linear speed given in mi/hr to the angular speed in rad/min.
First, let's convert the linear speed from miles per hour to inches per minute:
15 mi/hr * 5280 ft/mi * 12 in/ft * 1 hr/60 min = 15 * 5280 * 12 / 60 in/min
The distance traveled by the wheel in 1 minute will be the circumference of the wheel given its diameter:
Circumference = π * diameter = 3.14 * 24 in = 75.36 in
Now, we can find the number of revolutions the wheel makes in 1 minute:
Revolutions per minute = Distance traveled / Circumference
= (15 * 5280 * 12 / 60) / 75.36
= (15 * 5280 * 12) / (60 * 75.36)
= 15840 / 4521.6
≈ 3.502 revolutions per minute
a) The angular speed of the wheels in rad/min is the same as the number of revolutions per minute multiplied by 2π:
Angular speed in rad/min = Revolutions per minute * 2π
≈ 3.502 * 2π
≈ 22 rad/min
b) The wheels make approximately 3.502 revolutions per minute.
To find the angular speed of the wheels in rad/min, we need to convert the linear speed of the bicycle from mi/hr to ft/min and then divide it by the circumference of the wheel.
a) Convert the linear speed from mi/hr to ft/min:
1 mile = 5280 feet
1 hour = 60 minutes
So, 15 mi/hr = (15 * 5280) ft/(1 * 60) min = 1320 ft/min.
b) Find the circumference of the wheel:
The diameter of the wheel is given as 24 inches, which means the radius is 24/2 = 12 inches. To change it to feet, we divide by 12 because there are 12 inches in a foot. So, the radius is 12/12 = 1 foot.
The circumference of the wheel is given by the formula C = 2πr, where r is the radius. Hence, the circumference is C = 2π * 1 = 2π feet.
Now, divide the linear speed in ft/min by the circumference to get the angular speed in rad/min:
Angular speed = Linear speed / Circumference
Angular speed = 1320 ft/min / (2π ft) = 1320/ (2π) rad/min ≈ 209.773 rad/min.
Therefore, the angular speed of the wheels is approximately 209.773 rad/min.
To find the number of revolutions per minute, we divide the angular speed by 2π (since 2π radians = 1 revolution):
b) Number of revolutions per minute = Angular speed / (2π)
Number of revolutions per minute = 209.773 rad/min / (2π) ≈ 33.333 revolutions/min.
Therefore, the wheels make approximately 33.333 revolutions per minute.