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Circular Motion Unit 5
An axis is the straight line around which rotation takes place. When an object turns about an internal axis- that is, an axis located within the body of the object- the motion is called rotation.When an object turns about an external axis, the motion is called revolution.
Earth undergoes both types of rotational motion. It revolves around the sun once every 365 ¼ days and it rotates around an axis passing through its geographical poles once every 24 hours.
Linear speed Linear speed is the distance moved per unit of time. A point on the outer edge of a merry-go-round moves a greater distance in one complete rotation that a point near the center. The linear speed is greater on the outer edge of rotating objects than it is closer to its axis. The speed of something moving along a circular path can be called tangential speed because the direction of motion is always tangent to the circle.
Rotational speed Rotational speed is the number of rotations per unit time. All parts of the merry-go-round rotate about their axis in the same amount of time. Thus, all parts have the same rate of rotation. Rotational speed is expressed in revolutions per minute (RPM).
Tangential speed and rotational speed are related. The faster it turns, the faster your tangential speed is. Tangential speed is directly proportional to rotational speed and the radial distance from the axis of rotation. At the center of a rotating platform, right at its axis, you have no tangential speed at all, but you do have rotational speed. As you move away from the center, you move faster and faster- your tangential speed increases while your rotational speed stays the same.
Centripetal force • Any force that causes an object to follow a circular path • Means “center seeking” or “towards the center” • When an automobile rounds a corner, the sideways-acting friction between the tires and the road provides the centripetal force that holds the car on a curved path. If the force of friction is not great enough, the car fails to make the curve and the tires slide sideways. The car skids.
Centrifugal force • Outward force • Means “center fleeing” or “away from the center”
Suppose you are whirling a can. If the string on the whirling can breaks, it if often wrongly stated that centrifugal force pulls the can from its circular path. But in fact, when the string breaks the can goes off in a tangential straight-line path because no force acts on it.Now suppose there is a ladybug inside the whirling can. The can provides the centripetal force, not the centrifugal force necessary to hold the ladybug in a circular path. The “centrifugal-force effect” is attributed not to any real force but to inertia- the tendency of the moving body to follow a straight-line path
Our view of nature depends on the frame of reference from which we view it. For instance, when sitting on a fast-moving train, we have no speed at all relative to the train, but we have an appreciable speed relative to the reference frame of the ground outside.From a stationary frame outside the whirling can, we see there is no centrifugal force acting on the ladybug. We do see centripetal force acting on the can. In the rotating frame of reference of the whirling can, both centripetal force and centrifugal force act on the ladybug. However, centrifugal force is an effect of rotation. It is not part of an interaction so it cannot be a true force.
Gravity is simulated by centrifugal force. If the spinning can freely falls, the ladybug inside will experience a centrifugal force that feels like gravity when the can spins at an appropriate rate.
Today we live on the outer surface of a spherical planet, held here by gravity. In the years ahead many people will likely live in huge, lazily rotating space stations where centrifugal force simulates gravity. The simulated gravity will be provided so the people can function normally. Occupants in today’s space shuttles feel weightless because they lack a support force. They’re not pressed against a supporting floor by gravity, nor do they experience a centrifugal force due to spinning. But future space travelers need not be subject to weightlessness. Their space habitats will likely spin effectively supplying a support force and nicely simulating gravity.
The comfortable 1g we experience at Earth’s surface is due to gravity. Inside a rotating spaceship the acceleration experienced is the centripetal/ centrifugal acceleration due to rotation.
Formulas • Period: the time it takes for one full rotation or revolution of an object • Unit is seconds • Frequency: the number of rotations or revolutions per unit time • Unit is Hertz T= 1 f
Formulas When an object spins in a circle, the distance it travels in one revolution is the circumference of a circle. The time it takes is the period. speed= 2Πr T
formulas An object can move around in a circle with a constant speed yet still be accelerating because its direction is constantly changing. This is centripetal acceleration. centripetal acc= (linear speed)2 radius
Formulas If the mass is being accelerated towards the center of a circle, it must be acted upon by an unbalanced force that gives it this acceleration. centripetal force= mv2 r
After closing a deal with a client, Kent leans back in his swivel chair and spins around with a frequency of 0.5Hz. What is Kent’s period of spin?
Curtis’ favorite disco record has a scratch 12cm from the center that makes the record skip 45 times each minute. What is the linear speed of the scratch as it turns?
Missy’s favorite ride at the fair is the rotor, which has a radius of 4m. The ride takes 2s to make one full revolution.What is Missy’s linear speed on the rotor?What is Missy’s centripetal acceleration on the rotor?
Captain Chip, the pilot of a 60500kg jet plane, is told that he must remain in a holding pattern over the airport until it is his turn to land. If Captain Chip flies his plane in a circle whose radius is 50km once every 30 min, what centripetal force must the air exert against the wings to keep the plane moving in a circle?
Formula • Torque: a measurement of the tendency of a force to produce a rotation about an axis torque= perpendicular force x lever arm The lever arm is the distance from the pivot point, or fulcrum, to the point where the component of force perpendicular to the lever arm is being exerted. The longer the lever arm, the greater the torque. Keep in mind that when an object is balanced, all torques must also balance.
Ned tightens a bolt in his car engine by exerting 12N of force on his wrench at a distance of 0.40m from the fulcrum. How much torque must Ned produce to turn the bolt?
Mabel and Maude are seesawing on the school playground and decide to see if they can move to the correct location to make the seesaw balance. Mabel weighs 400N and she sits 2m from the fulcrum of the seesaw. Where should 450N Maude sit to balance the seesaw?
Moment of inertia:the resistance of an object to changes in its rotational motion Hoop rotating about its center: I= mr2 Hoop rotating about its diameter: I= (1/2)mr2 Solid cylinder: I= (1/2)mr2 Stick rotating about its center of gravity: I= (1/12)ml2 Stick rotating about its end: I= (1/3)ml2 Solid sphere rotating about its center of gravity: I= (2/5)mr2
Formula Angular momentum: the measure of how difficult it is to stop a rotating object Angular momentum (L)= (mass)(velocity)(radius)
On the Wheel of Fortune game show, a contestant spins that 15kg wheel that has a radius of 1.40m. What is the moment of inertia of this disk-shaped wheel?
Trish is twirling her 0.60m majorette’s baton that has a mass of 0.40kg. What is the moment of inertia of the baton as it spins about its center of gravity?
Jupiter orbits the sun with a speed of 2079m/s at an average distance of 71,398,000m. If Jupiter has a mass of 1.90x1027kg, what is its angular momentum as it orbits?
