Introduction to Uniform Circular Motion

Uniform Circular Motion An object moves at uniform speed in a circle of constant radius. Uniform circular motion is accelerated motion. Why?

When a ball on the end of a string is swung in a vertical circle, the ball is accelerating because A.the speed is changing. B.the direction is changing. C.the speed and the direction are changing. D.the ball is not accelerating. Checking Understanding Slide 6-13

Answer When a ball on the end of a string is swung in a vertical circle, the ball is accelerating because A.the speed is changing. B.the direction is changing. C.the speed and the direction are changing. D.the ball is not accelerating. Slide 6-14

Centrifugal Force It’s a myth! We need to go back to Newton’s Laws to properly explain the feeling you get on a merry-go-round or in a turning car.

When a car accelerates You, as a passenger, feel as if you are flung backward. Your inertia (mass) resists acceleration. You are NOT flung backward. Your body wants to remain at rest as the car accelerates forward.

When a car decelerates You, as a passenger, feel as if you are flung forward. Your inertia (mass) resists the negative acceleration. You are NOT flung forward. Your body wants to remain in motion at constant velocity as the car accelerates backwards.

When a car turns You feel as if you are flung to the outside. Your inertia resists acceleration. You are not flung out, your body simply wants to keep moving in straight line motion!

As a general rule Whenever you feel you are flung in a certain direction, you can bet the acceleration is pointing in the opposite direction. Remember the elevator problems? When you feel you are flying up, acceleration of the elevator is down. When you feel you are sinking down, acceleration is up.

Acceleration in Uniform Circular Motion The velocity vector at any given point is subjected to an acceleration that turns it, but does not speed it up or slow it down. The acceleration vector is always at right angles to the velocity. The acceleration points toward the center of the circle.

Acceleration in Uniform Circular Motion The acceleration responsible for uniform circular motion is referred to as centripetal acceleration.

When a ball on the end of a string is swung in a vertical circle: What is the direction of the acceleration of the ball? A.Tangent to the circle, in the direction of the ball’s motion B.Toward the center of the circle Checking Understanding Slide 6-15

Answer When a ball on the end of a string is swung in a vertical circle: What is the direction of the acceleration of the ball? A.Tangent to the circle, in the direction of the ball’s motion B.Toward the center of the circle Slide 6-16

Centripetal Acceleration a c = v 2 /r a c : centripetal acceleration in m/s 2 v: tangential speed in m/s r: radius in metersv V = 2πr / TT=period v acac v acac acac v Centripetal acceleration always points toward center of circle!

Force in Uniform Circular Motion A force responsible for uniform circular motion is referred to as a centripetal force. Centripetal force is simply mass times centripetal acceleration. F c = ma c

Centripetal Force F c = m a c F c = m v 2 / r F c : centripetal force in N v: tangential speed in m/s r: radius in meters v FcFc v FcFc FcFc v Always toward center of circle!

More on Centripetal Force Centripetal force is not a unique type of force. Centripetal forces always arise from other forces. You can always identify the real force which is causing the centripetal acceleration. Nearly any kind of force can act as a centripetal force.

Friction as centripetal force As a car makes a turn, the force of friction acting upon the turned wheels of the car provide the centripetal force required for circular motion.

Example Problem A level curve on a country road has a radius of 150 m. What is the maximum speed at which this curve can be safely negotiated on a rainy day when the coefficient of friction between the tires on a car and the road is 0.40? Slide 6-28

Tension as centripetal force As a bucket of water is tied to a string and spun in a circle, the force of tension acting upon the bucket provides the centripetal force required for circular motion.

Gravity as centripetal force As the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion.

Normal force as centripetal force An automobile turning on a banked curve uses the normal force to provide the necessary centripetal force.

Example Problem A curve on a racetrack of radius 70 m is banked at a 15 degree angle. At what speed can a car take this curve without assistance from friction?

Weight on a string moving in vertical circle Centripetal force arises from combination of tension and gravity.

Tennessee Tornado at Dollywood Centripetal force when you are upside down arises from a combination of normal force and gravity.

Centripetal Force can do no work A centripetal force alters the direction of the object without altering its speed. Since speed remains constant, kinetic energy remains constant, and work is zero.

The Universal Law of Gravity F g = Gm 1 m 2 /r 2 F g : Force due to gravity (N) G: Universal gravitational constant 6.67 x N m 2 /kg 2 m 1 and m 2 : the two masses (kg) r: the distance between the centers of the masses (m)

The Force of Gravity Slide 6-35

Acceleration due to gravity F g = mg = GmM E /r 2 What is g equivalent to? g = GM E /r 2

1) A typical bowling ball is spherical, weighs 16 kgs, and has a diameter of 8.5 m. Suppose two bowling balls are right next to each other in the rack. What is the gravitational force between the two—magnitude and direction? 2) What is the magnitude and direction of the force of gravity on a 60 kg person? (Mearth = 5.98x1024 kg, Rearth = 6.37 x 106 m) Example Problems Slide 6-36

Acceleration and distance

PlanetRadius(mMass (kg)g (m/s2) Mercury2.43 x x Venus6.073 x x Mars3.38 x x Jupiter6.98 x x Saturn5.82 x x Uranus2.35 x x Neptune2.27 x x Pluto1.15 x x

Kepler’s Laws 1. Planets orbit the sun in elliptical orbits. 2. Planets orbiting the sun carve out equal area triangles in equal times. 3. The planet’s year is related to its distance from the sun in a predictable way.

Kepler’s Laws

Satellites

Orbital Motion Gm e m s / r 2 = m e v 2 / r = The mass of the orbiting body does not affect the orbital motion!

Consider the see saw

The weight of each child is a downward force that causes the see saw to twist. The force is more effective at causing the twist if it is greater OR if it is further from the point of rotation.

Consider the see saw The twisting force, coupled with the distance from the point of rotation is called a torque.

What is Torque? Torque is a “twist” (whereas force is a push or pull). Torque is called “moment” by engineers. The larger the torque, the more easily it causes a system to twist.

Torque r F Hinge (rotates) Direction of rotation Consider a beam connected to a wall by a hinge. Now consider a force F on the beam that is applied a distance r from the hinge. What happens? A rotation occurs due to the combination of r and F. In this case, the direction is clockwise.

Torque  = F r sin   is torque is force r is “moment arm”  is angle between F and r r F  Hinge (rotates) Direction of rotation If we know the angle the force acts at, we can calculate torque!

Torque equation: simplified If  is 90 o …  = F r  is torque F is force r is “moment arm r F  Hinge: rotates Direction of rotation

We use torque every day Consider the door to the classroom. We use torque to open it. Identify the following: The point of rotation. The moment arm. The point of application of force.

Question Why is the doorknob far from the hinges of the door? Why is it not in the middle of the door? Or near the hinges?

Torque Units What are the SI units for torque? mN or Nm. Can you substitute Joule for Nm? No. Even though a Joule is a Nm, it is a scalar. Torque is a vector and cannot be ascribed energy units.

Now consider a balanced situation If the weights are equal, and the moment arms are equal, then the clockwise and counterclockwise torques are equal and no net rotation will occur. The kids can balance! 40 kg

Now consider a balanced situation  ccw =  cw This is called rotational equilibrium! 40 kg

Periodic Motion Repeats itself over a fixed and reproducible period of time. Mechanical devices that do this are known as oscillators.

An example of periodic motion… A weight attached to a spring which has been stretched and released. If you were to plot distance the vs time you would get a graph that resembled a sine or cosine function.

t(s) x(m) An example of periodic motion…

Simple Harmonic Motion (SHM) Periodic motion which can be described by a sine or cosine function. Springs and pendulums are common examples of Simple Harmonic Oscillators (SHOs).

Equilibrium The midpoint of the oscillation of a simple harmonic oscillator. Position of minimum potential energy and maximum kinetic energy.

All oscillators obey… Law of Conservation of Energy

Amplitude (A) How far the wave is from equilibrium at its maximum displacement. Waves with high amplitude have more energy than waves with low amplitude.

Period (T) The length of time it takes for one cycle of periodic motion to complete itself.

Frequency (f): How fast the oscillation is occurring. Frequency is inversely related to period. f = 1/T The units of frequency is the Herz (Hz) where 1 Hz = 1 s -1.

t(s) x(m) A T Parts of a Wave Equilibrium point

Springs A very common type of Simple Harmonic Oscillator. Our springs will be ideal springs. They are massless. They are compressible and extensible. They will follow a law known as Hooke’s Law.

Restoring force The restoring force is the secret behind simple harmonic motion. The force is always directed so as to push or pull the system back to its equilibrium (normal rest) position.

Hooke’s Law A restoring force directly proportional to displacement is responsible for the motion of a spring. F = -kx where F: restoring force k: force constant x: displacement from equilibrium

Hooke’s Law The force constant of a spring can be determined by attaching a weight and seeing how far it stretches. mgmg FsFs m F s = -kx

Hooke’s Law mmm x F x F F = -kx Spring compressed, restoring force out Spring at equilibrium, restoring force zero Spring stretched, restoring force in Equilibrium position

Period of a spring T = 2  m/k T: period (s) m: mass (kg) k: force constant (N/m)

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Pendulums The pendulum can be thought of as an oscillator. The displacement needs to be small for it to work properly.

Pendulum Forces T mg   mg sin 

Period of a pendulum T = 2  l/g T: period (s) l: length of string (m) g: gravitational acceleration (m/s 2 )