Circular Motion, Center of Gravity, & Rotational Mechanics Chapters 9, 10, & 11

Rotation and Revolution Axis – straight line around which rotation occurs Rotation – object turns about an internal axis (Earth rotates around its axis) Revolution – when an object turns about an external axis (Earth revolves around the sun)

Rotation

Revolution

Rotational Speed The linear speed of an object is greater near the outer edge of a rotating object than on the inner edge of the rotating object Tangential Speed – the speed of something moving along a circular path (the direction of motion is always tangent to the circle) Rotational Speed ( angular speed) – the number of rotations per unit of time (expressed in revolutions per minute or RPM) All parts of a rotating object rotate about their axis in the same amount of time!

Uniform Circular Motion

Tangential Speed ~ Radial Distance x Rotational Speed Tangential speed and rotational speed are related Tangential Speed ~ Radial Distance x Rotational Speed As you move away from the center of a rotating object, the tangential speed will increase while your rotational speed stays the same

Centripetal Force Centripetal Force – any force that causes an object to follow a circular path When a car goes around a corner, the friction between the tires and the road provides the centripetal force needed to keep the car going around the curve If not for the friction of the tires, the car would continue moving in the straight-line path

Centripetal Force

To Have or Not to Have Centripetal Force Without Centripetal Force With Centripetal Force

Centrifugal Force Centrifugal Force – the outward force associated with circular motion It is not a true force, but rather the effect that inertia tries to place on you as you follow a circular path From Newton’s First Law, the natural path of an object is a straight-line, the centripetal force is what keeps you going in a circle

Centrifugal Force

Center of Gravity Center of Gravity – point located at the object’s average position of weight For a symmetrical object, it is the geometric center of the object For an irregularly shaped object, there is more weight on end than the other, so the center of gravity is toward the heavier end Objects not made of the same material throughout (different densities), may have the center of gravity very far from the geometric center

Center of Gravity

Center of Mass Center of Mass – the average position of all the particles of mass that make up an object For almost all objects on or near Earth, center of gravity and center of mass are interchangeable If you threw an object in the air, you’d see it wobble around its center of gravity The sun wobbles also! As the planets move around the sun, they contribute to the overall center of mass of the solar system, so the sun wobbles off center This is how astronomers look for planets orbiting other stars!

Center of Mass in a Star System

Locating the Center of Gravity The center of gravity is the balance point, supporting that single point supports the whole object If you suspend any object at a single point, the center of gravity for that object will hang directly below (or at) the point of suspension The center of gravity may be located where no actual material exists (i.e. a ring)

Locating the Center of Gravity

Toppling If the CG of an object is above the area of support, it will remain upright If the CG extends outside the area of support, the object will topple The Leaning Tower of Pisa does not topple because its CG does not extend beyond its base

Toppling

Stability Unstable Equilibrium – an object balanced so that any displacement lowers its center of gravity Stable Equilibrium – an object balanced so that any displacement raises its center of gravity (requires work) Neutral Equilibrium – an object balanced so that its center of gravity is neither raised nor lowered with displacement

Stability

Center of Gravity of People When you stand upright with your arms hanging at your sides, your CG is within your body, typically 2 to 3 cm below your belly button The CG is slightly lower in women than in men, because women tend to be proportionally larger in the pelvis and smaller in the shoulders When you stand, your CG is somewhere above the support base of your feet, we spread them further apart in unstable situations (the bus) When you bend over to touch your toes, you are unconsciously extending the lower part of your body, putting your CG outside of your body (so you won’t topple over!)

Center of Gravity of People

Torque = force┴ x lever arm A torque is produced when a force is applied with “leverage” You use leverage when you use a screwdriver to open a can of paint The direction of your applied force is important, you would never try to open a door with a doorknob by push or pulling the doorknob sideways You apply your force PERPENDICULAR to the plane of the door Torque = force┴ x lever arm Greater torques are produced when both the force and lever arm are large

Torque

Torque and Center of Gravity If the direction of force is through the CG of the projectile, all the force can do is move the object as a whole; there will be no torque to turn the projectile If the force is directed “off center”, then in addition to motion of the CG, the projectile will rotate about its CG

Rotational Inertia An object rotating about an axis tends to keep rotating about that axis (look familiar?) Rotational Inertia – the resistance of an object to changes in its rotational motion A torque is required to change the rotational state of motion of an object Rotational inertia depends on the distribution of the mass of an object

Rotational Inertia

Rotational Inertia and Gymnastics Rotational inertia is least about the vertical head-to-toe axis (longitudinal) on any person, because most of the mass is concentrated there A rotation of your body along this axis is easiest The rotational inertia when your arms are extended is 3 times greater than when your arms are pulled in You rotate about your transverse axis when you do a flip or somersault The rotational inertia of a gymnast is up to 20 times greater when she is swinging in a fully extended position from a horizontal bar than after dismount when she somersaults in a tucked position (when she let goes and tucks, she is automatically increasing her rate of rotation by 20 times!)

Rotational Inertia and Gymnastics

Angular Momentum Angular Momentum – the “inertia of rotation” of rotating objects Like linear momentum, angular momentum is a vector quantity Rotational Velocity – when a direction is assigned to rotational speed Angular momentum = rotational inertia (I) x rotational velocity (ω) Angular momentum = mass (m) x velocity (v) x radius (r) Newton’s 1st Law can now be restated for angular momentum: An object or system of objects will maintain its angular momentum unless acted upon by an unbalanced external torque

Angular Momentum

Conservation of Angular Momentum The Law of Conservation of Angular Momentum: 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 at any other time

Conservation of Angular Momentum

Assignment Read Ch. 9-11 (pg. 122-164) Ch. 9: Do #31-38 (pg. 135), Appendix F #1-7 (pg. 674) Ch. 10: Do #21-34 (pg. 148-149) Ch. 11: Do #33-40 (pg. 167)