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Transcript
AP Physics – Little AP Test Review Helper – Gravity/Torque We studied the universal law of gravity and centripetal force together. Gravity Force that acts between two bodies that have mass. This force is given by the equation for gravity, which is called the universal law of gravity. FG Gm1m2 r2 N m G 6.67 x 1011 2 G is the universal gravity constant. It is given by: 2 kg Both bodies attract one another with the same magnitude force. A ball falls to the earth because of the force of gravity, so the ball falls. The earth also falls towards the ball, but the motion is so small that you don’t notice it. This is an inverse square function. Force of gravity decreases by the square of the distance. r is the distance between the center of mass for each of the objects. The acceleration of gravity on a planet can be found by setting the equation of weight, w mg , equal to the universal law of gravity. m1 g Gm1m2 r2 g Gm2 where m2 is the mass of the planet. r2 Of course, you remember Kepler’s 3 law of planetary motion: Planets move in ellipses with the sun at one focus. Planets sweep equal areas in equal times (speeding up in close to the sun.) R3/T2 is the same for all planets (later explained when Newton set the Gravity = Centripetal Force) Circular Motion: T 1 f Period: The time for one revolution. Frequency: How often a repeating event happens. For circular motion it would be measured in revolutions per second. Instantaneous velocity: Speed: f is frequency. The actual velocity (speed and direction) at any instant of time for a body that is traveling along a circular path. Its direction is tangent to its path. Also known as tangential velocity. Magnitude of instantaneous velocity. Also known as linear speed or tangential speed. The speed is equal to the circumference of the circle divided by the time to make one revolution. v x t 2 r T 1 Centripetal Acceleration: Equation provided for AP Test. v2 ac r Centripetal means “center seeking”. Direction is always towards center of circle. Centripetal Force: Force that changes direction of instantaneous velocity for a body that is traveling in a circle. Equation is not given, you have to derive it using centripetal acceleration and second law: v2 ac r F FNet ma v2 F m r mv 2 F r v ac ac ac v v Objects want to travel in straight line path – first law, but centripetal force changes the direction of the velocity, causing them to follow a circular path. Since object is pushed or pulled toward center, there is a feeling of being forced away from the center to the third law. This sensation is called the centrifugal force. Centrifugal force is a fictitious force. Centripetal force is always caused by some real force; tension in a string, friction, gravity, magnetic force, &tc. Problem gouge: 1. Think centripetal force if the object is going in a circle, accelerating towards the center, or changing direction. 2. In circular motion, the centripetal force is the sum of the forces acting on the object, so you can set it equal to the sum of all forces. FC F If one of the forces is a tension and the other force is the weight of the object, you get; FC t mg . You can then substitute for the centripetal force; FC t mg mv 2 t mg r You can then solve for whatever you want. 2 3. To find the minimum speed for an object to make a vertical circle, set the centripetal force equal to the weight of the object: mv2 mg r FC mg v rg 4. Finding the orbital velocity of a satellite. Set the centripetal force equal to the gravitational force. m2v 2 FC r v2 Gm1 r F G v m1m2 r2 Gm1 r mm m2v 2 G 1 2 r r2 so Gm r v so The mass is that of the body being orbited, like the sun for a planet or the earth for the moon, &tc. 5. Finding the orbital period of a satellite. Solve the velocity equation for time: v x t t x v x 2 r so t 2 r v Gm r The velocity of a satellite we already know - v 2 r t Gm r Torque t 2 2 2 r 2 Gm r 2 2 r 3 Gm 2 2 r 3 t Gm plug it in: r3 t 2 Gm rF sin The applied force that causes rotation: So there is excess torque in either the clockwise or counterclockwise direction. This will cause the object to rotate. 1. As always, ask what the object is doing. Is it rotating or is it standing still? 2. Set the direction of motion as positive. It will either rotate clockwise or counterclockwise. If you pick the wrong direction your final answer will be negative, telling you that you did thing in reverse. But, the answer will be correct nonetheless. If it is not moving pick one direction to be positive, it really doesn’t matter. But the other must be negative, so that the torque cancels. 3. Identify the sum of torque equation. clockwsise counterclockwsise or counterclockwsise clockwsise 3 4. Substitute the relevant force equations and solve (examples assume clockwise was positive direction) Rotating you get some + / Not Rotating rF sin clockwsise rF sin counterclockwsise 0 rF sin clockwsise rF sin counterclockwsise Angular momentum: Depends on mass (like regular momentum) and it also depends on mass distribution. As an ice skater brings their arms closer to the body they begin to spin faster, since the mass has a shorter distance to travel. Angular momentum is conserved. The radius gets smaller, but angular velocity increases (vice versa as the skater moves arms outward). A galaxy, solar system, star, or planet forms from a larger cloud of dust. As the cloud is pulled together by gravity its radius shrinks. So the angular velocity must increase. These objects all begin to spin faster and faster. That is why we have day and night. Rotational analogues And so on… t Angular acceleration (rad/s 2 ) 4