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Chapter 6: Lesson 2 - Centripetal Velocity and Acceleration
Chapter 6: Lesson 2 - Centripetal Velocity and Acceleration
In this lesson, we are moving onto objects traveling in a circle. Uniform circular motion is the motion of an object with a constant or uniform speed. When traveling in a circle, you are traveling around the perimeter of a circle, which is the circumference. The equation for the circumference of a circle is
c=2πr
When going around the circle, the time needed to make a complete revolution is called period. Period is abbreviated with a capital T, and the units are seconds. It is the time is takes the object to go around the circle ONCE!
The relationship between the circumference of a circle, the time to complete one cycle, and the speed of the object is an extension of the average speed equation, which we already know! v=d/t. The distance around the circle is 2πr and the time it takes to go around once is T.
v=(2πr)/T
The subscript c stands for centripetal, which we will talk about in lesson 3. You can also think of the subscript c as circular motion.
Revolution vs Rotation
Revolution vs Rotation
Revolution: When you move around something and your axis of rotation is outside your body, then you are revolving.
Rotation: When you spin around and your axis of rotation is inside your body, or internal, then you are rotating.
Revolution (an External Axis)
Revolution (an External Axis)
How fast an object moves when spinning in a circle depends on how far they are from the radius and how long it takes them to make one revolution.
Look at the two videos below. My family found a merry-go-round while camping in Rodgers City, MI in 2017. In the first video, my son Ryan is sitting on the outside edge of the merry-go-round. In the second video, he is sitting as close as he can to the axis of rotation. Play both videos at the same time and see where Ryan is moving the fastest, farther away from the axis of rotation on the outside edge or closer to the axis of rotation in the middle.
Revolution (external axis) - Large Radius
Revolution (external axis) - Large Radius
The farther you are away from the axis of rotation, the faster you will spin. Ryan is also traveling a farther distance every time he moves around the merry-go-round here than when he is sitting closer to the middle.
Revolution (external axis) - Small Radius
Revolution (external axis) - Small Radius
When Ryan is sitting close to the middle of the merry-go-round, the distance that he is traveling is not very far, but the time that it takes him to make one revolution is still the same time that it would take when he was sitting on the edge of the merry-go-round. Since he is not traveling as far, in the same amount of time, his rotational speed is much less when he is sitting in the center, then when he is sitting on the edge.
Rotation (an Internal Axis)
Rotation (an Internal Axis)
A figure skater can spin very quickly on the ice. When they do this, they are spinning with an internal axis of rotation. The axis of rotation goes right through their body. When they start spinning, they have their arms and legs far away from their body. As they spin and they pull their arms and legs closer to their body, more of their mass is closer to the axis of rotation. When their mass moves closer to the axis of rotation, they spin faster!
Demonstration Time with Mrs. Markey!
Demonstration Time with Mrs. Markey!
Mrs. Markey helps me demonstrate that when you are spinning with an internal axis of rotation and you bring more mass closer to the axis of rotation, you spin faster.
Watch the video and see what happens!
Circular Motion
Circular Motion
Remember that velocity is a vector quantity. The magnitude of a velocity vector is the instantaneous speed of the object. The direction of the velocity vector is the direction that the object is moving. When objects are moving in a circle, the speed of the object can stay the same, but the direction will constantly be changing.
The best word to describe the velocity of a vector is the word tangential. A tangent line is a line that touches the circle at one point but does not intersect it. The picture shows the direction of four different velocity vectors for an object moving clockwise.
When you move in a circle clockwise, you move in the same direction that the hands on a clock move. When you move counterclockwise, you move in the opposite direction that the hands on a clock move.
When talking about the speed of objects moving in a circle, you can also call this the tangential speed, because the velocity is at a tangent line to the circle.
Centripetal Acceleration
Centripetal Acceleration
Because the acceleration of an object moving in a circle is always in the direction of the net force acting on it, there must be a net force toward the center of the circle. This force can be provided by any number of agents.
Hammer throw, which is a track and field event, is when a hammer is hurled for distance, using two hands within a throwing circle. When a hammer thrower swings the hammer the force is the tension in the chain attached to the massive ball. The hammer thrower spins in a circle to increase the centripetal acceleration and then releases the hammer. The goal of the hammer throw is to get the hammer as far horizontally as you can. When the hammer is released, it travels in a tangent line to the circle that it was being spun around in.
Watch the video to see what the hammer throw looks like and to hear some physics behind the hammer throw.
Centripetal Velocity and Centripetal Acceleration
Centripetal Velocity and Centripetal Acceleration
When an object travels in a circle, there will always be a velocity and an acceleration vector. As stated above, the velocity vector will always point in a tangent line to the circle. In this case, the object is moving in a counterclockwise motion. You will need to know which direction the object is moving in order to find the velocity vector.
The acceleration of an object always points in the same direction as the net force. In lesson 3, we will learn that the centripetal force is directed towards the center of the circle. If there was not this force directed towards the center of the circle, the object would fly off in a tangent line because of Newton's First Law of Motion, an object in motion tends to stay in motion. The force pointing towards the center of the circle is what keeps the object in a circle.
The acceleration and velocity vectors are perpendicular to each other.
Look at this animation of a rollercoaster. Notice that the gravitational force ALWAYS points straight down and the normal force is always perpendicular to the surface of the track.
The shape of the track is called a Clothoid loop. This looks like an upside down tear drop. In a clothoid loop, the radius at the bottom of the loop is significantly larger than the radius at the top of the loop.
Building a rollercoaster with a clothoid loop allows the coaster to build up more speed going into and going out of the loop.
Check Your Understanding:
Check Your Understanding:
An object is moving in a clockwise direction around a circle at a constant speed. Use your understanding of the concepts of velocity and acceleartion to answer the next four questions. Use the diagrams below to help you answer the questions. Click on the down arrow when you have your answer to check to see if you are correct.
Which vector above (a, b, c, or d) represents the direction of the velocity vector when the object is located at point B on the circle?
The answer is d. The velocity vector points in a tangent line and since the object is moving clockwise, the velocity vector would be moving down at point B.
2. Which vector above (a, b, c, or d) represents the direction of the acceleration vector when the object is located at point C on the circle?
The answer is b. The acceleration vector always points towards the center of the circle.
3. Which vector above (a, b, c, or d) represents the direction of the acceleration vector when the object is located at point A on the circle?
The answer is d. The acceleration vector always points towards the center of the circle.
4. Which vector above (a, b, c, or d) represents the direction of the velocity vector when the object is located at point C on the circle?
The answer is a. The velocity vector points in a tangent line and since the object is moving clockwise, the velocity vector would be moving up and to the left at point C.
You have finished Lesson 2!
You have finished Lesson 2!
Be sure to head over to google classroom and fill out the exit pass.
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