1) A 6-ft propeller turns at 1200 revolutions per minute. Compare the angular velocity of the tip with the angular velocity at a point 10 in. from its center.
2) What is the angular velocity of a point on the tread of a 15-in. diameter tire on a car traveling 55 mph?
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To compare the angular velocity of the tip and a point 10 inches from the center of a 6-ft propeller turning at 1200 revolutions per minute, you first need to convert the measurements to the same units.
1) Convert 6 ft to inches:
6 ft * 12 inches/ft = 72 inches
2) Convert revolutions per minute to radians per second:
Angular velocity (rad/s) = (2π * revolutions) / (time in seconds)
Since there are 60 seconds in a minute:
Angular velocity of the tip = (2π * 1200) / 60 = 40π rad/s
Now, let's calculate the angular velocity at a point 10 inches from the center:
3) Convert 10 inches to feet:
10 inches / 12 inches/ft = 0.8333 ft
4) Calculate the radius of the point 10 inches from the center:
Radius = 6 ft / 2 + 0.8333 ft = 3.8333 ft
5) Calculate the speed at the point using the formula:
Angular velocity (rad/s) = (linear speed of the point) / (radius)
Linear speed of the point = angular velocity * radius
Angular velocity at the point = 40π rad/s * 3.8333 ft = 128π ft/s ≈ 402.1239 ft/s
Therefore, the angular velocity at the tip is 40π rad/s and the angular velocity at a point 10 inches from the center is approximately 402.1239 ft/s.
Now, let's calculate the angular velocity of a point on the tread of a 15-inch diameter tire on a car traveling at 55 mph:
1) Convert 55 mph to feet per second:
55 mph * 5280 ft/mile * (1/60) min/s * (1/60) hr/s = 80.6667 ft/s
2) Calculate the radius of the tire:
Radius = diameter / 2 = 15 inches / 2 = 7.5 inches = 0.625 ft
3) Calculate the angular velocity using the same formula as above:
Angular velocity (rad/s) = (linear speed of the point) / (radius)
Angular velocity of the point on the tread = 80.6667 ft/s / 0.625 ft = 129.0667 rad/s
Therefore, the angular velocity of a point on the tread of a 15-inch diameter tire on a car traveling 55 mph is approximately 129.0667 rad/s.
To compare the angular velocities in both scenarios, we need to first understand the relationship between angular velocity and linear velocity.
Angular velocity (ω) is defined as the rate at which an object rotates around an axis. It is usually measured in radians per second (rad/s).
Linear velocity (v), on the other hand, is the rate at which an object moves in a straight line. It is usually measured in units such as meters per second (m/s) or miles per hour (mph).
The relationship between angular velocity and linear velocity can be expressed as follows:
v = ω × r
where v is the linear velocity, ω is the angular velocity, and r is the distance from the axis of rotation to the point of interest.
Now, let's apply this concept to the given questions:
1) A 6-ft propeller turns at 1200 revolutions per minute. To compare the angular velocities, we need to consider two points: the tip of the propeller and a point 10 inches (0.83 ft) from its center.
Given:
Angular velocity at the tip = 1200 revolutions per minute
To calculate the angular velocity at the tip, we first need to convert the given angular velocity from revolutions per minute to radians per second. There are 2π radians in one revolution, and there are 60 seconds in one minute. So, we can convert the angular velocity as follows:
ω_tip = (1200 revolutions/minute) × (2π radians/rev) × (1 minute/60 seconds) = 40π radians/second
Now, we can use the relationship between angular velocity and linear velocity to calculate the linear velocity at the tip. Since the radius of the propeller is 6 ft, the distance from the center to the tip is also 6 ft. Therefore, the linear velocity at the tip is:
v_tip = ω_tip × r = (40π radians/second) × (6 ft) = 240π ft/second
To compare this with the angular velocity at a point 10 inches (0.83 ft) from the center, we need to calculate the radius from the center to that point:
r_point = 6 ft - 0.83 ft = 5.17 ft
Using the same relationship, the angular velocity at this point is:
ω_point = v_point / r_point
To find the linear velocity at this point, we need to make use of the angular velocity at the tip. Since the radius at this point is 5.17 ft, we have:
v_point = ω_tip × r_point = (40π radians/second) × (5.17 ft) = 206.8π ft/second
Therefore, the angular velocity at the point 10 inches from the center is:
ω_point = v_point / r_point = (206.8π ft/second) / (5.17 ft) = 40π radians/second
Comparing the angular velocities at the tip and the point 10 inches from the center, we find that they are equal:
ω_tip = ω_point = 40π radians/second
2) To calculate the angular velocity of a point on the tread of a 15-in. diameter tire on a car traveling 55 mph, we can follow a similar approach:
Given:
diameter of the tire = 15 inches
speed of the car = 55 mph
First, we need to convert the diameter of the tire from inches to feet:
diameter = 15 inches = 15 inches / 12 inches/foot = 1.25 feet
To calculate the radius of the tire, we divide the diameter by 2:
radius = 1.25 feet / 2 = 0.625 feet
Now, we can calculate the angular velocity using the relationship between linear velocity and angular velocity:
ω = v / r
To find the linear velocity, we need to convert the speed of the car from mph to feet per second:
speed = 55 mph × 5280 feet/mile × 1 hour/3600 seconds = 80.67 ft/second
Using the formula, we have:
ω = (80.67 ft/second) / (0.625 feet) = 129.07 rad/second
Therefore, the angular velocity of a point on the tread of the 15-in. diameter tire on a car traveling 55 mph is approximately 129.07 radians per second.