1

• The Conical Pendulum• As the ball revolves

faster, the angle increases• If the radius is 0.02 m

and the angle equal 30° • What’s the speed for a

given angle?

Example:

2

tansinLg

)sinLr(but

tanrgv

rg

vtan

then

)2........(....................mgcosT

)1.....(....................r

mvsinT

2

2

3

A worker drags(يجر) a crate along a rough, horizontal surface by pulling on a rope tied ربط) ) to the crate. the worker exerts a

force of 300 N on the rope that is inclined 37° to the horizontal .If the mass of the crate is 60 kg , and the coefficient

of kinetic friction is 0.3 , find the acceleration of the crate .

4

A 2-kg block is placed on top of a 5 – kg . A horizontal force of 40 N is applied to the 5- kg block.If the coefficient of kinetic friction between the 5-Kg and the surface is 0.2 , and assuming that the 2 –Kg block is in the verge of slipping ,a): What is the acceleration of the system .What is the coefficient of static friction is

5

• A 3kg block starts from rest at the top of 30º incline and slides a distance of 2m down the incline in 1.5s. Find

• (a) the acceleration of the block,

• (b) the coefficient of kinetic friction between the block and the plane,

• (c) the friction force acting on the block, (d) the speed of the block after it has slid 2m.

6

7

• Given m = 3kg,     θ = 30o,    x =2m,  

•   t = 1.5s

•            • Xo=0 then

• x = 1/2at 2   

•  2 = 1/2a (1.5)2   

•     a = 1.78 m/s2

•       

8

•        mg sin30 - f = ma •        f = m (g sin30 -a) f = 9.37N     •        N - mg cos30 = 0 •        N = mg cos30•    •     f = 9.37N•       µk = f / N = 0.368• V2=v02+2a(x-x0)

• v2 = 0 + 2(1.78)(2) = 7.11• then •        v = 2.67m/s

9

Problem: Forces are being applied to a box sitting on a surface with friction. Will the box move horizontally (along the surface)? F1=50N, F2=50N, Mass of the block 10kg, and µs=0.4. Find The Fn and Acceleration

Hint: must be greater or equal to xF sf

10

2

2s1

x

ss

nss

2n

2n

y

n

s/m4.1a

macosFfF

maF

N2.265.65x4.0Nuf

ORFuf

N5.65mgsinFF

0mgsinFF

0F

NF

11

Problem:  A 20 kg sled is being pulled across a horizontal surface at a constant velocity.  The pulling force has a magnitude of 80.0 N and is directed at an angle or 30.0° above the horizontal. 

Determine the coefficient of kinetic friction .

12

Centripetal ForceIf there is a centripetal acceleration, then the net

force must also be a centripetal force:

r

vmmaF cc

2

13

Tangential and Radial acceleration

14

Example: A car exhibits a constant acceleration of 0.300 m/s2 parallel to the roadway. The car passes over a rise in the roadway such that the top of the rise is shaped like a circle of radius 500 m. At the moment the car is at the top of the rise, its velocity vector is horizontal and has a magnitude of 6.00 m/s. What is the direction of the total acceleration vector for the car at this instant?

15

If the angle between

22

/072.0500

36sm

rar

22t

2r s/m309.0aaa

5.13tan 1

t

r

a

a

16

EX 3 A ball tied the end of string 0.5 m in length swings in a vertical circle under the influence of gravity . When the string makes an angle θ= 20 degre with the vertical , the ball has a speed of 1.5 m/s .

(a) Find the magnitude of the radial component of acceleration at this instant.(b) what is the magnitude of the tangential acceleration when θ= 20 degre.(c) find the magnitude and direction of the total acceleration a at

θ= 20 degre.

17

0

r

t1

22t

2r

tr

20t

2

22

r

375.4

4.3

a

atan

s/m6.5aaa

aaa

s/m4.320singa

s

m5.4

5.0

5.1

r

va

18

NOTE

19

20

• SOL

• Θ=19.136°

EX 1 :A Circular curve a road is designed for traffic moving at 60 km/hr without depending on the friction . If the radius of the curve is 80 m , what is the correct angle of the banking on

the road .

21

The moon revolves about earth in an orbit of radius

and makes on revolution in 27.3 days .Find the acceleration of the moon toward the

earth

km10x85.3 5

22

23

2

6

sm10x37.2

r

va

s/m1026T

r2v

S10x36.23600x24x3.27T

23

Example (a) Calculate the centripetal force exert on a 1000 kg car that negotiates a 600 m radius curve at 20.0 m/s. (b)

Assuming an unbanked curve, find the minimum static coefficient

of friction between the tires and the road.

24

A flat (unbanked ) curve on a highway has a radius of 100 m .if the coefficient of static –friction between the

tries ( العجلة . and the road is 0.2 (اطارWhat is the maximum speed with which the car will

have in order to round the curve successfully

25

.hr/km4.50gruv

mgN

r

vmNuf

s

2

ss

26

Problem: A particle moves in a circular path 0.4m in radius with constant speed.  If the particle makes five

revolution in each second of its motion, find (a) the speed of the particle and (b) its acceleration.

27

Since r=0.4m, t=0.4 s

sm56.125x512.2v

m512.24.0x14.3x2r2d

28

• Problem: A train slows down as it rounds a sharp horizontal turn, slowing from 90km/h to 50km/h in the 15s that it takes to round the bend.  The radius of the curve is 150m.  Compute the acceleration at the train.

29

30

Problem: I rotate a ball at an angle of 30o.  What is the centripetal acceleration?  If the string is 1 meter long, how fast is it rotating?

31

ProblemDriving in your car with a constant speed of 12 m/s, you encounter a bump in the road that has a circular cross section, as indicated in the Figure. If the radius of curvature of the bump is 35 m, find the apparent weight of a 67-kg person in your car as you pass over the top of the bump.

Nmg

a=v2/r