It's time to practice that golf swing in Episode 23! In order to keep track of objects as they rotate, physicists and mathematicians have created a unit known as a radian.
It's time to practice that golf swing in Episode 23! In order to keep track of objects as they rotate, physicists and mathematicians have created a unit known as a radian. (1:52) Do you know how a golfer can hit the ball faster with a driver than with an iron or a wedge? (4:42) Remember, rotating objects have displacements, velocities and accelerations of both the linear and angular varieties. (6:21)
The Question of the Day asks: (8:27)
If the radius is doubled for a rotating object, how does that change the angular velocity of the rotating object?
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Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on the new units of radians and other rotational quantities.
Let’s Zoom out:
Unit 7 – Torque and Rotational Motion
Topic - 7.1
Big Idea – Force Interactions
A putter, a sand wedge, and a driver are all types of clubs in a golfer’s set. Functionally, they are all similar. They are intended to allow users to rotate their bodies and extend their reach to hit an object. However, there are key differences in their design and usage. The driver is meant to hit the golf ball over the longest distances, and so the golfer needs to hit the ball at very fast speeds to achieve longer distances. The putter on the other hand is much shorter because it is intended for delicate applications that do not require the ball to be hit as fast.
Let’s Zoom in:
Before we begin, it should be explained that the point at the center of rotation for any object is known as the “axis of rotation” which you can treat just as you would the center of a circle. So, when we measure the radius, we are doing so from this location outward. We should also agree on a common convention for positive and negative directions of rotation. Generally, most agree that counterclockwise is the positive rotational direction.
In order to keep track of objects as they rotate, physicists and mathematicians have created a unit known as a radian. It isn’t really anything too special, it is just a radius distance around a circle. For example, the circumference of a circle can be found by the product of two, pi, and radius. Since two pi is a little bigger than 6, you can go a bit over six radius distances around a circle to complete a rotation. That is to say there are 2 pi radians or radius distances around any circle. Another way to relate the distance around a circle is using degrees. There are 360 degrees around, which is equal to 2 pi radians, so we can use a simple conversion factor to relate the concepts of degrees and radians. Using radians and degrees are ways to report rotational or angular displacements. Rather than measuring in meters around a circle, we use radians since it will specify how far around a circle you have traveled no matter the size, or radius, of the circle.
Something to note about radian measurements, they are technically unitless even though we write radians or “rad” after the quantities. Another way to look at radians is as a ratio of arc length (distance around a circle) to radius (distance from the center to the edge of the circle). Even though these are distances, meters divided by meters cancels the units. For example if a circle has a radius of 1 m, and you travel 4 m around, you will have traveled a little more than half way (pi radians) around the circle since you traveled 4 radians around... four radians, four...
FORE! How can a golfer hit the ball faster with a driver than with an iron or a wedge? A golfer can only rotate their body so fast, so in order to hit the golf ball far, they use a long club called a driver to increase their radius. The point that strikes the ball, the end of the club, is able to move more quickly. Imagine a meter stick that is held at one end, and allowed to rotate once around every second. We would say that it has an angular velocity of 1 rotation per second. Actually, we would say it rotates 2 Pi radians every second. Both of these are rotational quantities and are the same no matter what point of the meter stick you are referring to. In physics we use a capital Greek letter, “omega,” to represent angular velocity in equations, and we use radians per second as the unit. Angular velocity is different than the linear or tangential velocities we have dealt with until this point. Linear velocities look at the meters of displacement divided by the time, so a point on the end of the meter stick covers 2*pi*1 meter per second, while a point halfway down the meter stick will only travel 2*pi*0.5 meters per seconds or Pi meters per second. Golfers can take advantage of this relationship between radius and linear speed to hit a ball farther using a longer club, however they cannot do much to increase their maximum angular speed or how many radians they rotate in a second.
Since we have gotten displacement and velocity out of the way, we should at least mention that you can also have angular accelerations as well or rad/s2. Just like linear accelerations mean a change in linear velocity occurs, angular acceleration means that the number of radians rotated in a second is changing. An example of this would be with a lawn mower. When mowing the lawn, the blade rotates with some rate of radians per second, but when you power down the lawn mower, the angular speed of the blade decreases and eventually the blade comes to rest. I should remind you that this is now the third type of acceleration you have learned about. Linear acceleration, angular acceleration, and don’t forget centripetal acceleration. It is possible for a mass on the outside of a rotating wheel to have zero linear and angular acceleration if it is spinning at constant speed, but it will still have a centripetal acceleration since it is moving in a circle. This must be because the direction of the linear velocity is constantly changing, but the angular velocity and the linear speed are not.
If you are looking for a quick conversion between linear and angular quantities, a good rule of thumb is that you can take an angular quantity and multiply it by the radius to determine the linear quantity. This is true with displacement, velocity, and acceleration, but there is a notable exception to that rule that will be explained in a future episode, momentum.
To Recap…
Rotating objects have displacements, velocities and accelerations of both the linear and angular varieties. For the angular quantities we use radians, radians per second, and radians per second squared. It is easy to convert between linear and angular quantities by multiplying the angular value by the radius to find the linear quantity.
Coming up next on the APsolute RecAP Physics 1 Edition, we look at rotational kinematics and let the work from Unit 1 come around full circle.
Today’s Question of the Day focuses on angular velocity.
Question:
If the radius is doubled for a rotating object, how does that change the angular velocity of the rotating object?
a) angular velocity is doubled
b) angular velocity is halved
c) there is no change