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## Uniform Circular Motion

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**Uniform Circular Motion**Uniform Circular Motion**Uniform Circular Motion**• An object that moves at uniform speed in a circle of constant radiusis said to be in uniform circular motion. • Question: Why is uniform circular motion accelerated motion? • Answer: Although the speed is constant, the velocity is not constant since an object in uniform circular motion is continually changing direction.**Centrifugal Force**• Question: What is centrifugal force? • Answer: That’s easy. Centrifugal force is the force that flings an object in circular motion outward. Right? • Wrong!Centrifugal force is a myth! • There is no outward directed force in circular motion. To explain why this is the case, let’s review Newton’s 1st Law.**Newton’s 1st Law and cars**• When a car accelerates forward suddenly, you as a passenger feel as if you are flung backward. • You are in fact NOT flung backward. Your body’s inertia resists acceleration and wants to remain at rest as the car accelerates forward. • When a car brakes suddenly, you as a passenger feel as if you are flung forward. • You are NOT flung forward. Your body’s inertia resists acceleration and wants to remain at constant velocity as the car decelerates.**When a car turns**• You feel as if you are flung to the outside. You call this apparent, but nonexistent, force “centrifugal force”. • You are NOT flung to the outside. Your inertia resists the inward acceleration andyour body simply wants to keep moving in straight line motion! • As with all other types of acceleration, your body feels as if it is being flung in the opposite direction of the actual acceleration. The force on your body, and the resulting acceleration, actually point inward.**Centripetal Acceleration**• Centripetal(or center-seeking)acceleration points toward the center of the circle and keeps an object moving in circular motion. • This type of acceleration is at right angles to the velocity. • This type of acceleration doesn’t speed up an object, or slow it down, it just turns the object.**ac**v Centripetal Acceleration • ac = v2/r • ac: centripetal acceleration in m/s2 • v: tangential speed in m/s • r: radius in meters Centripetal acceleration always points toward center of circle!**Centripetal Force**• A force responsible for centripetal acceleration is referred to as a centripetal force. • Centripetal force is simply mass times centripetal acceleration. • Fc = m ac • Fc = m v2 / r • Fc: centripetal force in N • v: tangential speed in m/s • r: radius in meters Fc Always toward center of circle!**Any force can be centripetal**• The name “centripetal” can be applied to any force in situations when that force is causing an object to move in a circle. • You can identify the real force or combination of forces which are causing the centripetal acceleration. • Any kind of force can act as a centripetal force.**Static friction**As a car makes a turn on a flat road, what is the real identity of the centripetal force?**Tension**As a weight is tied to a string and spun in a circle, what is the real identity of the centripetal force?**Gravity**As the moon orbits the Earth, what is the real identity of the centripetal force?**Normal force with help from static friction**As a racecar turns on a banked curve on a racing track, what is the real identity of the centripetal force?**Tension,with some help from gravity**As you swing a mace in a vertical circle, what is the true identity of the centripetal force?**Gravity, with some help fromthe normal force**When you are riding the Tennessee Tornado at Dollywood, what is the real identity of the centripetal force when you are on a vertical loop?**Examples**mg T The maximum tension that a 0.50 m string can tolerate is 14 N. A 0.25-kg ball attached to this string is being whirled in a vertical circle. What is the maximum speed the ball can have (a) the top of the circle, (b)at the bottom of the circle?**Examples**At the bottom? T mg**Highway Curves**• As long as the tires do not slip, the friction is static. If the tires do start to slip, the friction is kinetic, which is bad in two ways. • The kinetic frictional force is smaller than the static • The static frictional force can point towards the center of the circle, but the kinetic frictional force opposes the direction of motion making it very difficult to regain control of the car and continue around the curve.**Not enough friction….**• If the frictional force is sufficient then Ff = Fc • If the frictional force is insufficient, the car will tend to move more nearly in a straight line**Sample problem**• A 1200-kg car rounds a corner of radius r = 45 m. If the coefficient of static friction between tires and the road is 0.93 and the coefficient of kinetic friction between tires and the road is 0.75, what is the maximum velocity the car can have without skidding?**Banked Curves**• Banking the curve can help keep cars from skidding. • For every banked curve there is one speed where the entire centripetal force is supplied by the horizontal component of the normal force and no friction is required**Sample problem**You whirl a 2.0 kg stone in a horizontal circle about your head. The rope attached to the stone is 1.5 m long. a) What is the tension in the rope? (The rope makes a 10o angle with the horizontal). b) How fast is the stone moving?**Torque**• A torque is an action that causes objects to rotate. • Torque is notthe same thing as force. • For rotational motion, the torqueis what is most directly related to the motion, not the force.**Torque**• Motion in which an entire object moves is called translation. • Motion in which an object spins is called rotation. • The point or line about which an object turns is its center of rotation. • An object can rotate and translate.**Torque**• Torque is created when the line of action of a force does not pass through the center of rotation. • The line of action is an imaginary line that follows the direction of a force and passes though its point of application.**Torque**Lever arm length (m) t = r x F Torque (N.m) Force (N)**9.1 Calculate a torque**• A force of 50 newtons is applied to a wrench that is 30 centimeters long. • Calculate the torque if the force is applied perpendicular to the wrench so the lever arm is 30 cm.**Rotational Equilibrium**• When an object is in rotational equilibrium, the net torque applied to it is zero. • Rotational equilibrium is often used to determine unknown forces. • What are the forces (FA, FB) holding the bridge up at either end?**Calculate using equilibrium**• A boy and his cat sit on a seesaw. • The cat has a mass of 4 kg and sits 2 m from the center of rotation. • If the boy has a mass of 50 kg, where should he sit so that the see-saw will balance?**Calculate a torque**• It takes 50 newtons to loosen the bolt when the force is applied perpendicular to the wrench. • How much force would it take if the force was applied at a 30-degree angle from perpendicular? • A 20-centimeter wrench is used to loosen a bolt. • The force is applied 0.20 m from the bolt.**Simple Harmonic Motion**AP Physics B**Simple Harmonic Motion**Back and forth motion that is caused by a force that is directly proportional to the displacement. The displacement centers around an equilibrium position.**Springs – Hooke’s Law**One of the simplest type of simple harmonic motion is called Hooke's Law. This is primarily in reference to SPRINGS. The negative sign only tells us that “F” is what is called a RESTORING FORCE, in that it works in the OPPOSITE direction of the displacement.**Hooke’s Law**Common formulas which are set equal to Hooke's law are N.S.L. and weight**Example**A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount? 110 N 1000 N/m**Hooke’s Law from a Graphical Point of View**Suppose we had the following data: k =120 N/m**We have seen F vs. x Before!!!!**Work or ENERGY = FDx Since WORK or ENERGY is the AREA, we must get some type of energy when we compress or elongate the spring. This energy is the AREA under the line! Area = ELASTIC POTENTIAL ENERGY Since we STORE energy when the spring is compressed and elongated it classifies itself as a “type” of POTENTIAL ENERGY, Us. In this case, it is called ELASTIC POTENTIAL ENERGY.**Elastic Potential Energy**The graph of F vs.x for a spring that is IDEAL in nature will always produce a line with a positive linear slope. Thus the area under the line will always be represented as a triangle. NOTE: Keep in mind that this can be applied to WORK or can be conserved with any other type of energy.**Example**A slingshot consists of a light leather cup, containing a stone, that is pulled back against 2 rubber bands. It takes a force of 30 N to stretch the bands 1.0 cm (a) What is the potential energy stored in the bands when a 50.0 g stone is placed in the cup and pulled back 0.20 m from the equilibrium position? (b) With what speed does it leave the slingshot? 3000 N/m 300 J 109.54 m/s**Springs are like Waves and Circles**The amplitude, A, of a wave is the same as the displacement ,x, of a spring. Both are in meters. CREST Equilibrium Line Period, T, is the time for one revolution or in the case of springs the time for ONE COMPLETE oscillation (One crest and trough). Oscillations could also be called vibrations and cycles. In the wave above we have 1.75 cycles or waves or vibrations or oscillations. Trough Ts=sec/cycle. Let’s assume that the wave crosses the equilibrium line in one second intervals. T =3.5 seconds/1.75 cycles. T = 2 sec.**Frequency**The FREQUENCY of a wave is the inverse of the PERIOD. That means that the frequency is the #cycles per sec. The commonly used unit is HERTZ(HZ).**SHM and Uniform Circular Motion**Springs and Waves behave very similar to objects that move in circles. The radius of the circle is symbolic of the displacement, x, of a spring or the amplitude, A, of a wave.**SHM and Uniform Circular Motion**• The radius of a circle is symbolic of the amplitude of a wave. • Energy is conserved as the elastic potential energy in a spring can be converted into kinetic energy. Once again the displacement of a spring is symbolic of the amplitude of a wave • Since BOTH algebraic expressions have the ratio of the Amplitude to the velocity we can set them equal to each other. • This derives the PERIOD of a SPRING.**Example**A 200 g mass is attached to a spring and executes simple harmonic motion with a period of 0.25 s If the total energy of the system is 2.0 J, find the (a) force constant of the spring (b) the amplitude of the motion 126.3 N/m 0.18 m**Pendulums**Pendulums, like springs, oscillate back and forth exhibiting simple harmonic behavior. A shadow projector would show a pendulum moving in synchronization with a circle. Here, the angular amplitude is equal to the radius of a circle.**Pendulums**Consider the FBD for a pendulum. Here we have the weight and tension. Even though the weight isn’t at an angle let’s draw an axis along the tension. q mgcosq q mgsinq**Pendulums**What is x? It is the amplitude! In the picture to the left, it represents the chord from where it was released to the bottom of the swing (equilibrium position).