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Extension of Circular Motion & Newtons Laws

Extension of Circular Motion & Newtons LawsChapter 6Mrs. WarrenKings High School

SReview from Chapter 4Uniform Circular MotionCentripetal Acceleration

Two analysis models using Newtons Laws of Motion have been developed.The models have been applied to linear motion.Newtons Laws can be applied to other situations: Objects traveling in circular pathsMotion observed from an accelerating frame of referenceMotion of an object through a viscous mediumMany examples will be used to illustrate the application of Newtons Laws to a variety of new circumstances.A particle moves with a constant speed in a circular path of radius r with an acceleration.The magnitude of the acceleration is given by

The centripetal acceleration, , is directed toward the center of the circle.The centripetal acceleration is always perpendicular to the velocity.

2Uniform Circular Motion, ForceA force is associated with the centripetal acceleration.The force is also directed toward the center of the circle.

Applying Newtons Second Law along the radial direction gives

Section 6.1

Uniform Circular Motion, cont.A force causing a centripetal acceleration acts toward the center of the circle.It causes a change in the direction of the velocity vector.If the force vanishes, the object would move in a straight-line path tangent to the circle.See various release points in the active figureSection 6.1

A small ball of mass, m, is suspended from a string of length, L. The ball revolves with constant speed, v, in a horizontal circle of radius, r. Find an expression for v in terms of the geometry.

The Conical Pendulum

Bc the string sweeps out the surface of a cone, the system is known as a conical pendulum. The object is in equilibrium in the vertical direction .It undergoes uniform circular motion in the horizontal direction.Fy = 0 T cos = mgFx = T sin = m acv is independent of m

5The Conical PendulumA puck of mass 0.500 kg is attached to the end of a cord. The puck moves in a horizontal circle of radius 1.50 m. If the cord can withstand a maximum tension of 50.0 N, what is the maximum speed at which the puck can move before the cord breaks? Assume the string remains horizontal during the motion.

The speed at which the object moves depends on the mass of the object and the tension in the cord.The centripetal force is supplied by the tension.

The maximum speed corresponds to the maximum tension the string can withstand.

6The Flat CurveModel the car as a particle in uniform circular motion in the horizontal direction.Model the car as a particle in equilibrium in the vertical direction.The force of static friction supplies the centripetal force.The maximum speed at which the car can negotiate the curve is:

Note, this does not depend on the mass of the car.

How did we get this formula? 7The Flat CurveA 1500 kg car moving on a flat, horizontal road negotiates a curve as shown on the previous slide. If the radius of the curve is 35.0 m and the coefficient of static friction between the tires and dry pavement is 0.523, find the maximum speed the car can have and still make the turn successfully. The Banked RoadwayThese are designed with friction equaling zero.Model the car as a particle in equilibrium in the vertical direction.Model the car as a particle in uniform circular motion in the horizontal direction.There is a component of the normal force that supplies the centripetal force.The angle of bank is found from

How do we derive that formula??

The banking angle is independent of the mass of the vehicle.If the car rounds the curve at less than the design speed, friction is necessary to keep it from sliding down the bank.If the car rounds the curve at more than the design speed, friction is necessary to keep it from sliding up the bank.

9The Banked CurveA civil engineer wishes to redesign the curved roadway in such a way that a car will not have to rely on friction to round the curve without skidding. In other words, a car moving at the designated speed can negotiate the curve even when the road is covered with ice. Such a road is usually banked, which means that the roadway is tilted toward the inside of the curve. Suppose the designated speed for the road is to be 30.0 mi/h and the radius of the curve is 35.0 m. At what angle should the curve be banked?13.4 m/s

10Ferris WheelA child of mass m rides on a Ferris wheel as shown. The child moves in a vertical circle of radius 10.0 m at a constant speed of 3.00 m/s. Determine the force exerted by the seat on the child at the bottom of the ride. Express your answer in terms of the weight of the child, mg. Do the same when the child is at the top of the ride.

The normal and gravitational forces act in opposite direction at the top and bottom of the path.Categorize the problem as uniform circular motion with the addition of gravity.The child is the particle.At the bottom of the loop, the upward force (the normal) experienced by the object is greater than its weight.At the top of the circle, the force exerted on the object is less than its weight.

11Non-Uniform Circular MotionThe acceleration and force have tangential components. produces the centripetal acceleration produces the tangential accelerationThe total force is

Section 6.2

12Vertical Circle with Non-Uniform SpeedA small sphere of mass, m, is attached to the end of a cord of length R and set into motion in a vertical circle about a fixed point O as illustrated. Determine the tangential acceleration of the sphere and the tension in the cord at any instant when the speed of the sphere is v and the cord makes an angle with the vertical.

Section 6.2

The gravitational force exerts a tangential force on the object.Look at the components of FgModel the sphere as a particle under a net force and moving in a circular path.Not uniform circular motion

The tension at any point can be found13Motion in Accelerated FramesA fictitious force results from an accelerated frame of reference.The fictitious force is due to observations made in an accelerated frame.A fictitious force appears to act on an object in the same way as a real force, but you cannot identify a second object for the fictitious force.Remember that real forces are always interactions between two objects.Simple fictitious forces appear to act in the direction opposite that of the acceleration of the non-inertial frame.

Section 6.3Centrifugal Force

From the frame of the passenger (b), a force appears to push her toward the door.From the frame of the Earth, the car applies a leftward force on the passenger.The outward force is often called a centrifugal force.It is a fictitious force due to the centripetal acceleration associated with the cars change in direction.In actuality, friction supplies the force to allow the passenger to move with the car.If the frictional force is not large enough, the passenger continues on her initial path according to Newtons First Law.

15Coriolis Force

This is an apparent force caused by changing the radial position of an object in a rotating coordinate system.The result of the rotation is the curved path of the thrown ball.From the catchers point of view, a sideways force caused the ball to follow a curved path.

Although fictitious forces are not real forces, they can have real effects.Examples:Objects in the car do slideYou feel pushed to the outside of a rotating platformThe Coriolis force is responsible for the rotation of weather systems, including hurricanes, and ocean currents.

16Motion with Resistive ForcesThe medium exerts a resistive force, , on an object moving through the medium.The magnitude of. .The direction of isThe magnitude of can depend on the speed in complex ways.We will discuss only two: is proportional to vGood approximation for is proportional to v2Good approximation for

Section 6.4

The magnitude of depends on the medium.The direction of is opposite the direction of motion of the object relative to the medium.This direction may or may not be in the direction opposite the objects velocity according to the observer . nearly always increases with increasing speed.The magnitude of can depend on the speed in complex ways.We will discuss only two: is proportional to vGood approximation for slow motions or small objects is proportional to v2Good approximation for large objects

17Resistive Force Proportional To SpeedThe resistive force can be expressed as

b depends on the property of the medium, and on the shape and dimensions of the object.The negative sign indicates is in the opposite direction to .

Section 6.4Resistive Force Proportional To Speed, ExampleAssume a small sphere of mass m is released from rest in a liquid.Forces acting on it are:Resistive forceGravitational forceAnalyzing the motion results in

Section 6.4

19Resistive Force Proportional To Speed, Example, cont.Initially, v = 0 and dv/dt = gAs t increases, R increases and a decreasesThe acceleration approaches 0 when R mgAt this point, v approaches the terminal speed of the object.

Terminal SpeedTo find the terminal speed, let a = 0

Solving the differential equation gives

t is the time constant andt = m/b

Section 6.4

Terminal SpeedA small sphere of mass 2.00 g is released from rest in a large vessel filled with oil, where it experiences a resistive force proportional to its speed. The sphere reaches a terminal speed of 5.00 cm/s. Determine the time constant and the time at which the sphere reaches 90.0% of its terminal speed. Resistive Force Proportional To v2For objects moving at high speeds through air, the resistive