Rotational Motion

Average speed = Circumference ÷ time Uniform circular motion is the motion of an object in a circle with a constant or uniform speed. If a car is traveling in a circle at 5 m/s, that means it will cover 5 meters along the perimeter of a circle every second. The distance of one complete cycle around the perimeter of a circle is known as the circumference. This relationship between the circumference of a circle, the time to complete one cycle around the circle, and the speed of the object is merely an extension of the average speed equation. Average speed = Circumference ÷ time The circumference of a circle is given by the equation: C = 2πr

Average Speed = 2πr ÷ time Doing a substitution you get the following equation: Average Speed = 2πr ÷ time Where “r” is the radius of the circle. Objects moving with uniform circular motion are traveling at constant speed, but they do not have a constant velocity. Since velocity involves speed and direction something moving in a circle is constantly changing direction. This means an object moving in a circle is constantly accelerating. The direction of the velocity vector at any instant is in the direction of a tangent line drawn to the circle at the object's location.

When an object is rotating it has two different kinds of speed When an object is rotating it has two different kinds of speed. There is the linear speed. Linear speed is speed as we already know it: the distance covered divided by the time. The other is called rotational speed. Rotational speed is the number of rotations per unit of time. A common unit of rotational speed is rpm’s or rotations per minute. Old phonograph records were distinguished by the rotational speed needed to play them. A single was also called a “45” because it required the record player to spin at 45 rpm to play it. The LP or long playing record, also called an album, required 33 1/3 rpm to play them. There were older records which required 78 rpm’s to play.

Toy tops like this can spin at rotational speeds of 3000 rpm. Linear speed is also called tangential velocity because its vector is directed along a tangent line to the circle of its motion. Toy tops like this can spin at rotational speeds of 3000 rpm.

The rotational speed of a spinning object is the same everywhere on that object. Pretend this circle is a spinning carousel with a radius “R”. No matter where you stand on the carousel you will spin at the same rate. Your number of rpm’s will not change. What about linear speed? Is someone standing near the center of the carousel going faster, slower, or the same as someone standing near the edge? R

The person near the center is going slower than someone near the edge The person near the center is going slower than someone near the edge. Linear speed is the distance divided by the time. Who is traveling the greater distance? The one near the edge is traveling a greater distance over the same period of time. Therefore, the one near the edge has a greater linear speed. To summarize, the farther away from the center the greater the linear speed. But the rotational speed is constant everywhere on the spinning object.

Newton’s first law stated that an object at rest remained at rest or an object in motion remained in motion unless affected by an outside force. This property of resistance to change in motion was called inertia. This property exists for rotating objects as well. An object at rest will resist being rotated or an object already rotating will resist any change to that rotation unless acted upon by some outside influence. This is called rotational inertia. Similar to linear inertia, rotational inertia depends on the mass of the object. However, rotational inertia differs from linear inertia in that it also depends on the distribution of mass.

The farther away mass is from the axis of rotation the greater the rotational inertia is. That means the more difficult it would be to start it rotating. Of course, this also means that once an object with a large rotational inertia is rotating it is more difficult to stop it. The balancing pole carried by a tightrope walker increases the acrobat’s rotational inertia thus making it easier for him regain balance.

Rotating around center. Rotating around diameter. Rotational inertia is affected by the shape of the object as well as around which axis it is rotating. Consider two identical hoops. One hoop is rotating around its center axis. The other is spinning around its diameter. Rotating around center. Rotating around diameter.

Which way is it easier to rotate a hoop, around its center or around its diameter. The rotational inertia (I) to rotate a hoop around its center axis is given by the equation: I = mr2 “m” is for mass and “r” is the radius of the hoop. To rotate the same hoop around its diameter is given by the equation: I = ½mr2. The rotational inertia is less when spinning around the diameter, so it is easier to spin a hoop around its diameter.

The rotational inertia of an object depends on the object’s shape as much as its mass. The shape determines the distribution of mass and how far the axis of rotation is from the center of mass. Page 130 in the textbook has the various equations for rotational inertias of different objects.

Torque = Force x lever arm Torque is the rotational counterpart of force. A force will make a stationary object move in a straight line. A torque will make a stationary object rotate around an axis. A torque is created when you apply a force to an object a certain distance from the axis of rotation. The distance the applied force is from the axis of rotation is called the lever arm because it provides leverage. The torque on an object is the force times the lever arm. Torque = Force x lever arm

This seesaw is balanced because the torque provided by the man is equal in magnitude, but opposite in direction to the torque provided by the rocks. The net torque on the seesaw is zero; therefore, no rotation.

The man’s weight wants to rotate the seesaw in a clockwise direction The man’s weight wants to rotate the seesaw in a clockwise direction. The weight of the rocks wants to rotate the seesaw in a counterclockwise direction. Since the rocks and the man are the same distance away from the axis of rotation of the seesaw, they must have the same weight to produce equal torques. What would happen if the man moved closer to the center of the seesaw? The lever arm would be less; therefore, his torque would be less and the seesaw would rotate counterclockwise. The side with the rocks would fall. How could you balance the seesaw again?

You could move all the rocks closer to the center of the seesaw You could move all the rocks closer to the center of the seesaw. You could also remove some of the rocks. Since the man’s torque is less, removing some of the rocks will reduce the torque on the other side. If you use a wrench to turn a stubborn bolt you want to do what you can to increase the torque. Torque increases by increasing the force applied or the lever arm. If you can apply the force from a longer distance the torque increases. This is why the longer the wrench the easier it is to turn the bolt. The applied force must be perpendicular to the lever arm. If you tug on the wrench with your arm at an angle it is more difficult to turn the bolt.

The center of mass is the average position of all the mass that makes up an object. For a symmetrical object like a baseball, marble, or a cylinder, the center of mass is at the geometric center of the object. C M C M

For objects that are not symmetrical, like a baseball bat, the center of mass will be located in the heavier section of the object. The center of mass of the bat is in the wider end because there is more mass distributed there than in the smaller end.

When an object like a baseball bat is thrown it appears to wobble, but it is rotating around its center of gravity. (The terms center of mass and center of gravity are interchangeable.) The center of gravity itself follows a smooth parabolic path. It is possible for the center of mass of an object to be where no physical mass exits. Objects like a doughnut, boomerang, or an empty cup have a center of mass where no mass actually exists.

Center of mass where no mass exists.

The location of the center of mass is important for determining stability. If you drop a straight line through the center of mass of any object, and it falls within the base of the object; the object is said to be in stable equilibrium. This means it will not topple over by its own weight. It is possible for an object to be balanced, but the smallest force would cause the center of mass to be lowered. Such an object is said to be in unstable equilibrium. Consider a cone balanced on its point: The center of mass is directly over the base but the slightest force will lower the center of mass, and it will topple.

A balanced object in which any force does not change the position of the center of mass is said to be in neutral equilibrium. A cone lying on its side is in neutral equilibrium because no matter what force is exerted on the cone, the center of mass stays the same distance above its base. The cone will just roll in a circle.

If an object’s center of gravity(mass)remains over a base of support the object will not topple. So part of an object can extend over the edge of a stable without falling as long as its center of gravity is positioned over the table. Consider a kicked football. A football in flight often tumbles end over end this is because the kicker’s foot struck the ball beneath the ball’s center of gravity. What about a football which is not tumbling during flight? This occurs when the kicker’s foot strikes the ball directly at its center of gravity.

Any force directed toward the center of motion is called centripetal force. The term “centripetal” means “center-seeking” or towards the center. To twirl a can on the end of string, it becomes necessary to keep pulling on the string to keep the moving in a circle. The tension in the string provides the centripetal force. The moon is held in a near circular orbit around the earth because the gravitational force between the earth and the moon is toward the center of the earth-moon system. Therefore, the gravitational force is also centripetal force. The same can be said for the planets and the sun.

When a car is turning around a corner, the friction between the tires and the road provides the centripetal force. This is evidenced by the fact that if a car tries to turn on an icy road, the reduced friction of the ice does not provide enough force to turn, and the car continues moving in a straight line despite the wheels being in a turning position. Many amusement park rides depend on centripetal force. An old-fashioned roller coaster stays on the tracks even when upside-down going through a loop due to centripetal force provided by the tracks on the wheels.

Centrifugal force is a “fake” force Centrifugal force is a “fake” force. Centrifugal means “center-fleeing” or “away from the center”. Taking the example of the can twirling on the end of a string, if the string broke the can would fly away from the string. Centrifugal force is used to the describe the “force” that is moving the can away from the broken string. This is a fallacy. The can is really moving away from the broken string because of the absence of force. The centripetal force that was being provided by the string is gone. Due to the law of inertia, the can will continue in the path it would be moving without the centripetal force, which is a straight line.

It just so happens that the straight line path tangent to the point on the circle exactly where the string broke. Path of can after string breaks.

If living things were placed inside a rotating structure, the centrifugal force caused by the rotation would feel like gravitational force. With the proper radius and proper rotational velocity, the rotation can be set to simulate Earth’s gravity. So it will be possible, someday, to build a space station in which the inhabitants would be able to move around as if they were walking on Earth. The problem is such a structure would have to be very large; therefore, they would need the ability to be spun at very high speeds. Such technology does not yet exist. The International Space Station is a large structure, but it does not rotate.

Inhabitants of the International Space Station are living in “weightless” conditions and are subject to the side effects of long term weightless existence. The most common of these side effects are loss of muscle strength and loss of bone calcium.

Angular momentum = m x v x r Just like the way objects moving in a straight line have linear momentum, objects that are rotating have angular momentum along the axis of rotation. The angular momentum of a bicycle wheel is directed along the wheel’s axle. The angular momentum is the rotational inertia times the rotational velocity. The angular momentum is also mass x velocity x radius. Angular momentum = m x v x r Like linear momentum, angular momentum is also conserved. Angular momentum can only be changed if the object is affected by an outside torque.

The conservation of angular momentum is the concept involved when you see an ice skater spin faster just by bringing her hands closer to his/her body. By bringing their hands closer together they reduce their radius and thus their rotational velocity must increase to conserve angular momentum.