Chapter 8 Rotational Motion

Objectives Apply the law of conservation of angular momentum to a system where no net external torque acts. Determine the change in angular velocity of a system where the moment of inertia of the objects that make up the system changes.

Exploration Lab Spinning Bicycle Wheel Lab What Can Set You Spinning?

Angular Momentum Newton’s first law of inertia for rotating systems states that an object or system of objects will maintain its angular momentum unless acted upon by an unbalanced external torque.

Anything that rotates keeps on rotating until something stops it. Angular Momentum Anything that rotates keeps on rotating until something stops it. Angular momentum is defined as the product of rotational inertia, I, and rotational velocity, . angular momentum = rotational inertia × rotational velocity () = I × 

Angular Momentum Like linear momentum, angular momentum is a vector quantity and has direction as well as magnitude. When a direction is assigned to rotational speed, we call it rotational velocity. Rotational velocity is a vector whose magnitude is the rotational speed.

Angular Momentum Angular momentum depends on rotational velocity and rotational inertia.

Angular Momentum The operation of a gyroscope relies on the vector nature of angular momentum.

Angular Momentum For the case of an object that is small compared with the radial distance to its axis of rotation, the angular momentum is simply equal to the magnitude of its linear momentum, mv, multiplied by the radial distance, r. angular momentum = mvr This applies to a tin can swinging from a long string or a planet orbiting in a circle around the sun. Video on changing factors mvr

Angular Momentum An object of concentrated mass m whirling in a circular path of radius r with a speed v has angular momentum mvr.

Angular Momentum An external net force is required to change the linear momentum of an object. An external net torque is required to change the angular momentum of an object.

8-8 Angular Momentum and Its Conservation In analogy with linear momentum, we can define angular momentum L: We can then write the total torque as being the rate of change of angular momentum. If the net torque on an object is zero, the total angular momentum is constant. (8-18)

Angular Momentum It is easier to balance on a moving bicycle than on one at rest. The spinning wheels have angular momentum. When our center of gravity is not above a point of support, a slight torque is produced. When the wheels are at rest, we fall over. When the bicycle is moving, the wheels have angular momentum, and a greater torque is required to change the direction of the angular momentum.

Angular Momentum The lightweight wheels on racing bikes have less angular momentum than those on recreational bikes, so it takes less effort to get them turning.

Conservation of Angular Momentum Angular momentum is conserved when no external torque acts on an object.

Conservation of Angular Momentum Angular momentum is conserved for systems in rotation. The law of conservation of angular momentum states that if no unbalanced external torque acts on a rotating system, the angular momentum of that system is constant. With no external torque, the product of rotational inertia and rotational velocity at one time will be the same as at any other time.

Demo Rotating disk and books What happens to rotational inertia when books are brought into the body. Rotational inertia decreases.

Conservation of Angular Momentum When the man pulls his arms and the whirling weights inward, he decreases his rotational inertia, and his rotational speed correspondingly increases. Video

Conservation of Angular Momentum The man stands on a low-friction turntable with weights extended. Because of the extended weights his overall rotational inertia is relatively large in this position. As he slowly turns, his angular momentum is the product of his rotational inertia and rotational velocity. When he pulls the weights inward, his overall rotational inertia is decreased. His rotational speed increases! Whenever a rotating body contracts, its rotational speed increases.

8-8 Angular Momentum and Its Conservation Therefore, systems that can change their rotational inertia through internal forces will also change their rate of rotation: video

Period 5 Conservation of Angular Momentum Rotational speed is controlled by variations in the body’s rotational inertia as angular momentum is conserved during a forward somersault. This is done by moving some part of the body toward or away from the axis of rotation. video

Conservation of Angular Momentum A falling cat is able to execute a twist and land upright even if it has no initial angular momentum. During the maneuver the total angular momentum remains zero. When it is over, the cat is not turning. This cat rotates its body through an angle, but does not create continuing rotation, which would violate angular momentum conservation. video

Period 1 Conservation of Angular Momentum Although the cat is dropped upside down, it is able to rotate so it can land on its feet.

Physics of Falling Cats How does a cat land on its legs when dropped? … Moment of inertia is important ... To understand how a cat can land on it's feet, you must first know some concepts of rotational motion, since the cat rotates as it falls. Reminder: The moment of inertia of an object is determined by the distance it's mass is distributed from the rotational axis. Relating this to the cat, if the cat stretches out it's legs and tail, it increases it's moment of inertia; conversely, it can decrease it's moment of inertia by curling up. Remember how it was proved by extending your professor’s arms while spinning around on a swivel chair? Just as a more massive object requires more force to move, an object with a greater moment of inertia requires more torque to spin. Therefore by manipulating it's moment of inertia, by extending and retracting its legs and rotating its tail, the cat can change the speed at which it rotates, giving it control over which part of it's body comes in contact with the ground.

Physics of Falling Cats ... and the conservation of angular momentum ... If a cat is dropped they almost always tend to land on their feet because they use the conservation of angular momentum to change their orientation When a cat falls, as you would expect, its centre of mass follows a parabolic path. The cat falls with a definite angular momentum about an axis through the cat’s centre of mass. When the cat is in the air, no net external torque acts on it about its centre of mass, so the angular momentum about the cat’s centre of mass cannot change. By pulling in its legs, cat can considerably reduce it rotational inertia about the same axis and thus considerably increase its angular speed. Stretching out its legs increases its rotational inertia and thus slows the cat’s angular speed. Conservation of angular momentum allows cat to rotate its body and slow its rate of rotation enough so that it lands on its feet

Conservation of Angular Momentum Falling cat twists different parts of its body in different directions so that it lands feet first At all times during this process the angular momentum of the cat as a whole is zero A free-falling cat cannot alter its total angular momentum. Nonetheless, by swinging its tail and twisting its body to alter its moment of inertia, the cat can manage to alter its orientation

Falling Cats: More Information How does a cat land on its legs when dropped? Cats have the seemingly unique ability to orient themselves in a fall allowing them to avoid many injuries. This ability is attributed to two significant feline characteristics: “righting reflex” and unique skeletal structure. The “righting reflex” is the cat’s ability to first, know up from down, and then the innate nature to rotate in mid air to orient the body so its feet face downward. Animal experts say that this instinct is observable in kittens as young as three to four weeks, and is fully developed by the age of seven weeks. A cat’s “righting reflex” is augmented by an unusually flexible backbone and the absence of a collarbone in the skeleton. Combined, these factors allow for amazing flexibility and upper body rotation. By turning the head and forefeet, the rest of the body naturally follows and cat is able reorient itself. Like many small animals, cats are said to have a non-fatal terminal falling velocity. That is, because of their very low body volume-to-weight ratio these animals are able to slow their decent by spreading out (flying squirrel style). Animals with these characteristics are fluffy and have a high drag coefficient giving them a greater chance of surviving these falls.

8-9 Vector Nature of Angular Quantities The angular velocity vector points along the axis of rotation; its direction is found using a right hand rule:

Angular Momentum of a Rigid Body

8-9 Vector Nature of Angular Quantities Angular acceleration and angular momentum vectors also point along the axis of rotation.

Exploration with Gyroscopes Explain how they work using Physics concepts. Video

Exploration with Gyroscopes What conclusions did you come up with? Gyroscopic Precession Explain how they work using Physics concepts.

Anti-Gravity Wheel Anti-Gravity Wheel Explained

Practice Problem 1 (#52) What is the angular momentum of a 2.8 kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? How much torque is required to stop it in 6.0s?

Practice Problem 1 (#52) answer What is the angular momentum of a 2.8 kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? How much torque is required to stop it in 6.0s?

Practice Problem 2 (similar to 54) A diver (such as the one shown in Fig. 8-28) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes two rotations in 1.5s when in the tuck position, what is her angular speed (rev/s) when in the straight position?

Practice Problem 2 A diver (such as the one shown in Fig. 8-28) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes two rotations in 1.5s when in the tuck position, what is her angular speed (rev/s) when in the straight position? This is a conservation of angular momentum problem.  The angular kinetic energy would not be conserved as the diver would do work in tucking that would be manifested as rotational kinetic energy.  (it would go up) Now, the formula for angular momentum is: L = I So basically, L before = L after: I1 = I2 Let I1 be the moment of inertia when she is straight, and I2 be the moment in the tucked position: I1 = (3.5)I2  = (2 rev)/(1.5 s) = 1.3333 rev/s (3.5)I2 = I2(1.3333 rev/s) (3.5) = (1.3333 rev/s)  = (1.3333 rev/s)/(3.5) = 0.38 rev/s

Practice Problem 3 (#55) A figure skater during her final can increase her rotation rate from an initial rate of 1.0 rev every 2.0s to a final rate of 3.0rev/s. If her initial moment of inertia was 4.6kg.m2 , what is her final moment of inertia? How does she physically accomplish this change?

Practice Problem 3 (#55) A figure skater during her final can increase her rotation rate from an initial rate of 1.0 rev every 2.0s to a final rate of 3.0rev/s. If her initial moment of inertia was 4.6kg.m2 , what is her final moment of inertia? How does she physically accomplish this change? This is a conservation of angular momentum problem. Now, the formula for angular momentum is: L = I So basically, L before = L after: I1 = I2  = (1 rev)/(2 s) = .5 rev/s = 3.0 rev/s I1 = 4.6 kgm2 (4.6 kgm2)(.5 rev/s) = I23.0 rev/s I2 = 0.77 kgm2 (Note that the units cancel, so we don't have to convert to radians per second) Skaters can change their moment of inertia by pulling in their arms and legs closer tot he axis of rotation

Summary of Chapter 8 Angles are measured in radians; a whole circle is 2π radians. Angular velocity is the rate of change of angular position. Angular acceleration is the rate of change of angular velocity. The angular velocity and acceleration can be related to the linear velocity and acceleration. The frequency is the number of full revolutions per second; the period is the inverse of the frequency.

Summary of Chapter 8, cont. The equations for rotational motion with constant angular acceleration have the same form as those for linear motion with constant acceleration. Torque is the product of force and lever arm. The rotational inertia depends not only on the mass of an object but also on the way its mass is distributed around the axis of rotation. The angular acceleration is proportional to the torque and inversely proportional to the rotational inertia.

Summary of Chapter 8, cont. An object that is rotating has rotational kinetic energy. If it is translating as well, the translational kinetic energy must be added to the rotational to find the total kinetic energy. Angular momentum is If the net torque on an object is zero, its angular momentum does not change.

Homework Chapter 8 problems #51 and 53 Test on chapter 8 Tuesday

Closure Kahoot 8-8