The Kit Carson County Carousek makes 3 revolutions per minute.
A.) Find the linear velocity in feet per second of someone riding a horse that is 22.5 feet from the center.
B.) The linear velocity of the person on the inside of the carousel is 3.1 feet per second. How far is the person from the center of the carousel?
C.) How much faster is the rider on the outside going than the rider on the inside?
w=3*2PI/60 rad/sec
velocity= radius*w
same equation
subtract the two velocities...
To solve these problems, we need to use the relationship between angular velocity, linear velocity, and radius of rotation.
The formula we'll use is:
Linear velocity = Angular velocity * Radius
where:
- Linear velocity is the speed of an object moving along a circular path,
- Angular velocity is the rate at which an object rotates, measured in radians per unit of time, and
- Radius is the distance from the center of rotation to the object.
A.) Find the linear velocity in feet per second of someone riding a horse that is 22.5 feet from the center.
In this case, we're given that the carousel makes 3 revolutions (i.e., 2π radians) per minute. We need to convert the angular velocity into radians per second to match the units.
Angular velocity = (3 revolutions / 1 minute) * (2π radians / 1 revolution) * (1 minute / 60 seconds)
Angular velocity = 3π / 10 radians per second
Now, use the formula mentioned earlier:
Linear velocity = Angular velocity * Radius
Linear velocity = (3π / 10 radians per second) * 22.5 feet
Linear velocity ≈ 70.7 feet per second
So, the linear velocity of someone riding a horse that is 22.5 feet from the center is approximately 70.7 feet per second.
B.) The linear velocity of the person on the inside of the carousel is 3.1 feet per second. How far is the person from the center of the carousel?
We're given the linear velocity and we need to find the radius.
First, rearrange the formula:
Linear velocity = Angular velocity * Radius
Radius = Linear velocity / Angular velocity
Plug in the given values:
Radius = 3.1 feet per second / (3π / 10 radians per second)
Radius = (3.1 * 10) / (3π)
Radius ≈ 3.27 feet
So, the person on the inside of the carousel is approximately 3.27 feet from the center.
C.) How much faster is the rider on the outside going than the rider on the inside?
To find the speed difference, we'll subtract the linear velocity of the rider on the inside from the linear velocity of the rider on the outside.
The linear velocity of the rider on the outside is already given as 70.7 feet per second.
The linear velocity of the rider on the inside is given as 3.1 feet per second.
Speed difference = Linear velocity outside - Linear velocity inside
Speed difference = 70.7 feet per second - 3.1 feet per second
Speed difference ≈ 67.6 feet per second
So, the rider on the outside is going approximately 67.6 feet per second faster than the rider on the inside.