Physics 111Lecture 5

Circular MotionDr. Ali ÖVGÜN

EMU Physics Department

www.aovgun.com

January 22, 2017

Multiple Objects

q A block of mass m1 on a rough, horizontal surface is connected to a ball of mass m2 by a lightweight cord over a lightweight, frictionless pulley as shown in figure. A force of magnitude F at an angle θ with the horizontal is applied to the block as shown and the block slides to the right. The coefficient of kinetic friction between the block and surface is μk. Find the magnitude of acceleration of the two objects.

January 22, 2017

Multiple Objectsq m1:

q m2:amamgmTF yy 222 ==−=∑

0sin

cos

1

11

=−+=

==−−=

∑∑

gmFNF

amamTfFF

y

xkx

θ

θ

θsin1 FgmN −=

2 1cos ( )kF f m a g m aθ − − + =

2

1 2

cos kF f m gam mθ − −=

+

)(2 gamT +=

January 22, 2017

Uniform circular motion

Constant speed, or,constant magnitude of velocity

Motion along a circle:Changing direction of velocity

Uniform Circular Motion: Definition

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Uniform Circular Motion: Observations

q Object moving along a curved path with constant speedn Magnitude of velocity: samen Direction of velocity: changingn Velocity : changingn Acceleration is NOT zero!n Net force acting on an

object is NOT zeron “Centripetal force” amFnet

!!=

v!

January 22, 2017

January 22, 2017

Uniform Circular Motionq Velocity:

n Magnitude: constant vn The direction of the velocity is

tangent to the circleq Acceleration:

n Magnitude: n directed toward the center of

the circle of motionq Period:

n time interval required for one complete revolution of the particle

rvac2

=

rvac2

=

vrT π2=

vac!! ⊥

January 22, 2017

Centripetal Forceq Acceleration:

n Magnitude: n Direction: toward the center of

the circle of motionq Force:

n Start from Newton’s 2nd Law

n Magnitude:

n Direction: toward the center of the circle of motion

rvac2

=

rvac2

=

amFnet!!

=

vac!! ⊥

rmvmaF cnet

2

==

netFr

netFr

netFr

vFnet!!

⊥

netc Fa!! ||

January 22, 2017

What provide Centripetal Force ?

q Centripetal force is not a new kind of forceq Centripetal force stands for any force that keeps

an object following a circular path

q Centripetal force is a combination of n Gravitational force mg: downward to the groundn Normal force N: perpendicular to the surfacen Tension force T: along the cord and away from objectn Static friction force: fs

max = µsN

rmvmaF cc

2

==

January 22, 2017

a

What provide Centripetal Force ?

rmvT

maTFnet2

=

==

rvmmgN

mamgNFnet2

+=

=−=

mg

N

January 22, 2017

Problem Solving Strategyq Draw a free body diagram, showing and labeling all

the forces acting on the object(s)q Choose a coordinate system that has one axis

perpendicular to the circular path and the other axis tangent to the circular path

q Find the net force toward the center of the circular path (this is the force that causes the centripetal acceleration, FC)

q Use Newton’s second lawn The directions will be radial, normal, and tangentialn The acceleration in the radial direction will be the centripetal

accelerationq Solve for the unknown(s)

January 22, 2017

Example 1:The Conical Pendulumq 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 and a

mg

T θ

44.0tan

4.0sin

sin

cos

0cos 2 5 5

22

2

=−

=

==

===

=

=−====

∑

∑

rLr

Lr

rmvmaTF

mgTmgTF

mrmLkgm

cx

y

θ

θ

θ

θθ

January 22, 2017

The Conical Pendulum

44.0tan

4.0sin

sin

cos

0cos 2 5 5

22

2

=−

=

==

==

=

=−====

∑

∑

rLr

Lr

rmvTF

mgTmgTF

mrmLkgm

x

y

θ

θ

θ

θθ

22

2

2

m/s 3.4tan

m/s 9.2tansin

tan

tan

cos

sin

===

==

=

=

=

=

θ

θθ

θ

θ

θ

θ

grva

Lgv

rgv

grvmgTrmvT

q Find v and a

Example 2: The Effect of Speed on Centripetal Force

q The model airplane has a mass of 0.90 kg and moves at constant speed on a circle that is parallel to the ground. The path of the airplane and the guideline lie in the same horizontal plane because the weight of the plane is balanced by the lift generated by its wings. Find the tension in the 17 m guideline for a speed of 19 m/s.

January 22, 2017

q In vertical circular motion the gravitational force must also be considered. An example of vertical circular motion is thevertical “loop-the-loop” motorcycle stunt. Normally, the motorcycle speed will varyaround the loop.

q The normal force, FN, and the weight of the cycle and rider, mg, are shown at fourlocations around the loop.

January 22, 2017

Vertical Circular Motion

January 22, 2017

qThere is a minimum speed the rider must have at point 3 in order to stay on the loop.

qThis speed may be found by settingin the centripetal force equation or point 3, i.e. in

January 22, 2017

January 22, 2017

Problem 1:

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Problem 2:

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Problem 3:

Problem 4:

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January 22, 2017

Problem 5:

Problem 6:

January 22, 2017

Problem 7: