Centripetal Acceleration & Angular Momentum. Physics - 4 th Six Weeks

Transcription

1 Centripetal Acceleration & Angular Momentum Physics - 4 th Six Weeks

2 Centripetal Force and Acceleration Centripetal Acceleration (A C ) is the acceleration of an object towards the center of a curved or circular path. Centripetal comes from the Latin for center seeking For a ball to accelerate towards the center and not follow Newton s st Law, a force must be acting upon it. The force is Centripetal Force, (F C ) which is the force caused by the grip or pull between the object and the surface it is upon causing it to be pulled towards the center of the curve. Centripetal Force is the result of another force such as gravity, tension, friction, the normal force, or the electrostatic force between electrically charged objects.

3 Centrifugal Force v Centripetal Force Historically, and in everyday speech people talk of the centrifugal force The Centrifugal force is the force that pulls a rotating or spinning object out from the center. Centrifugal comes from the Latin for center fleeing Devices like the washing machine, machines that separate out substances, or those that allow pilots to experience extreme forces in a lab are called centrifuges

4 Centrifugal Force v Centripetal Force However, there really isn t a centrifugal force. The apparent Centrifugal Force is caused by the object s inertia. If centripetal force were removed, the object would actually travel in a straight line (how it had been moving) due to its inertia Remember, that according to Newton s st Law of Motion the object will take a straight path at a constant velocity unless acted upon by an outside unbalanced force. The Centripetal Force is that force If there were a Centrifugal force pulling away, when centripetal force were removed an object would move straight out and travel in the direction of the centrifugal force. V is the inertia, whereas the curve is the resulting path

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6 Calculating Centripetal Acceleration Note: Tangential Velocity is how fast a point on a circular object (or an object moving in a circular path) is moving a certain distance from the radius of a circle.

7 Pro-tip: for convenience, you could put the formula into a circle diagram! Common units: v t = m/s a c =m/s 2 r = m v t 2 v t 2 = tangential velocity a c = centripetal acceleration r = radius a c r Note: in the formula that tangential velocity is squared or v t 2 Therefore: when solving for tangential velocity there will be an additional step of using the square root function ( a c r) to get rid of the square.

8 Example Problem # An amusement park ride spins at 3 m/s. The ride has a radius of 4 m and the ride takes 2 seconds to make a complete revolution. What is the centripetal acceleration? Given: radius = 4 m tangential velocity = 3 m/s Not needed time = 2 sec a c = a c = a c = (3m s )2 4 m 69 m2 s 2 4 m 69 m2 s 2 4 m a c = m/s 2

9 Example Problem #2 Jimmy moves around in a merry go round at 2 m/s. If the radius of the merry go round was 4 m, what was the centripetal acceleration? Given: radius = 4 m tangential velocity = 2 m/s a c = (2m s )2 4 m a c = 4 m2 s 2 4 m a c = 4 m2 s 2 4 m a c = m/s 2

10 Example Problem #3 A ball is twirled in a circle with a radius of 2 m. If the centripetal acceleration was 0.5 m/s 2, what was the tangential velocity? Given: radius = 2 m centripetal acceleration = 0.5 m/s 2 V t = a c x r V t = V t = m/s 0.5 m/s 2 x 2 m

11 Example Problem #4 Centripetal Jimmy has a mass of 80 kg and moves around in a merry go round at.5 m/s. If the radius of the merry go round was 4 m, what centripetal force did he experience? Force F c = mv2 r F c = 80 kg (.5 m s )2 4 m F c = 80 kg x 2.25 m2 s 2 4 m F c = 80 kg x m s 2 4 F c = 45 kg x m/s 2 Remember: N = kg x m/s 2 F c = 45 N

12 Angular Momentum (aka Rotational Momentum) Recall that anything that is moving has momentum Momentum is the quantity of motion, and if something is moving in a straight line it has Linear Momentum Therefore, anything that is moving at an angle (or around a curve or in a circle) has Angular Momentum The SI unit for angular momentum is kg x m 2 /s

13 Conservation of Angular Momentum Remember that according to the Law of Conservation of Momentum, unless acted upon by an outside force, the momentum of a system is conserved. So it stands to reason, that unless acted upon by an outside force, a rotating object will maintain a constant amount of rotational or angular momentum. This is known as the Law of Conservation of Angular Momentum A spinning top would continue spinning as it is forever if it weren t for gravity providing a torque, and friction slowing it down

14 Rotational Inertia (aka The Moment of Inertia) Inertia is the resistance to change. The measure of the resistance to change in rotation, or the resistance to torque is Rotational Inertia Rotational Inertia is more often called the Moment of Inertia Rotational Inertia is a product of an object s mass multiplied by its radius squared. The Moment of Inertia is depicted in a formula as a capital letter I (for Inertia) The SI unit is kg * m 2 Rotational Inertia of an airplane affects how the steering forces control pitch, roll, and yaw.

15 Angular Momentum footballs and accuracy In football (or in shooting a gun) a spiral has a benefit in that while it may go a bit slower (since some energy that would push it forward is used to rotate it), it does have great rotational inertia and momentum. Accuracy improves since greater inertia and momentum allows it to plow through the air without much loss of velocity due to air resistance. This improvement in accuracy is what lead to a much greater loss of life in the American Civil War compared to previous wars with guns that lacked rifled barrels

16 Rotational Inertia (Moment of Inertia) Recall that the resistance to torque or rotation is the Moment of Inertia The longer something is, the harder it is to rotate with a force (or to change its rotation) It is very difficult to move something so long A longer moment of inertia will move something with less of a moment of inertia This is why tight rope walkers use long poles to steady themselves

17 Rotational Inertia (aka The Moment of Inertia) Conservation of angular momentum explains the angular acceleration of a figure skater as she brings in her outstretched arms or when a diver moves from a straight position to a tucked position during a dive. Figure skaters reduce their moments of inertia by pulling in their arms which allows them to spin faster Moment of Inertia & Angular Velocity are indirectly related to one another Angular Velocity, or how quickly the angle changes (essentially the speed going around & around) is represented as ω or lower case omega ω is in radians per second. For those nongeometry types, radian is about 57.3 degrees, so rad/s = 57.3 /s and revolution = 6.3 radians

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19 Spinning Wheels and Balancing a Bicycle: the basics You are aware that if you stand on one foot you will have to hop & twist around to stay upright. You are having to adjust your center of mass to stay over your base (your foot) A bicyclist makes tiny adjustments by steering to do the same to keep CG over the wheels A moving bicycle or motorcycle, with spinning wheels has angular or rotational momentum It is harder for an outside torque (such as gravity) to change the movement of a moving, spinning motorcycle or bicycle. In a nutshell, once the wheels are lined up a certain way they want to stay lined up a certain way. You can supply an internal force as the rider to change that, but it is very difficult for an outside force to do so. When a cyclist is stopped at a red light they often will steer back and forth, side to side a bit to keep CG above the two points of contact to stay balanced Though the exact physics is even more complex

20 Circle Diagrams for the Angular Momentum formulas L I I ω m r 2 I = Rotational Inertia (Moment of Inertia) L = Angular Momentum ω = Angular Velocity m = mass r = radius

21 Rotational Inertia = Mass x radius 2 I = m x r 2 Key Formulas Also, Rotational Inertia = Angular Momentum / Angular Velocity I = L ω Angular Velocity = Angular Momentum/ Rotational Inertia ω = L I Rotational (or angular) Momentum = Rotational Inertia x Angular Velocity L = I x ω Radius = the square root of rotational inertia / mass r = I m Mass = rotational inertia / the radius squared m = I r 2 Units: Rotational Inertia = kg x m 2 Angular Velocity = rad/s Angular Momentum = kg x m 2 /s Mass = kg Radius = meters

22 Example # : Rotational Inertia A figure skater has a mass of 60 kg and her arms extended making her radius 0.8 m. What was her rotational inertia? Rotational Inertia = Mass x (radius 2 ) I = 60 kg x (0.8 m) 2 I = 60 kg x 0.64 m 2 I = 38.4 kg x m 2

23 Example # 2: angular velocity A diver had a angular momentum of 50 kg x m 2 /s, what was their angular velocity if their rotational inertia was.25 kg x m 2? ω = L I ω = 50 kg x m2 /s.25 kg x m/s ω = 50 kg x m2 /s.25 kg x m2 Note: the radians are understood FYI radian = 57.3 degrees so this is also /sec or 6.37 turns/sec ω = 50 kg x m2 /s ω = 50 kg x m2 /s ω = 40 rad/sec.25 kg x m2.25 kg x m 2

24 Ex # 3: Rotational Inertia (Moment of Inertia) A toy spun with an angular inertia of 2 kg x m 2 /s, what was its rotational inertia if its angular velocity was 20 rad/s? I = L ω I = 2 kg x m2 /s 20 rad/s I = 2 kg x m2 /s 20 rad/sec Note: the radians are technically unit less to begin with so unit wise they are ignored I = 2 kg x m2 /s 20 rad/sec I = 0. kg x m 2

25 Example # 4: Rotational Momentum A ball has a rotational momentum of 5 kg x m2/s and an angular velocity of 2 rad/s. What was its rotational momentum? Rotational Momentum = I x ω L = 5 kg x m 2 x 2 rad s L = 60 kg x m 2 /s Note: Radians are technically unit less so they are ignored in cancelling units

26 Example # 5: mass An box was spinning on a Lazy Susan. It had a moment of inertia of 4 kg x m 2, what was the mass of the box if the radius of the spin was 0.5 m? m = I r 2 m = 4 kg x m2 0.5 m 2 m = 4 kg x m2 0.5 m2 m = 4 kg x m2 0.5 m 2 m = 6 kg

27 Example # 6: radius A device was spinning. It had a mass of 2 kg and an angular inertia of 8 kg x m 2. What was the radius of the spin? Essentially r = I m r 2 = I m r 2 = r 2 = r 2 = 8 kg x m2 2 kg 8 kg x m2 r 2 = 9 m 2 8 kg x m2 2 kg 2 kg r = 9 m 2 r = 3 m