Dynamics Lecture

DYNAMICS OF RIGID BODIES Motion of a Particle Particle is a term used to denote an object of point size. A system of  particles which formed into appreciable size is termed as body. These terms may may appl apply y equa equall lly y to the the same same obje object ct.. The The eart earth h for for examp xample le may may be assumed as a particle in comparison with its orbit, whereas to an observer on the earth, it is a body with appreciable size. In eneral, a particle is an object whose size is so small in comparison to the size of its path. I.

Kinematics

1) Rectilinear Rectilinear Translation Translation (Motion (Motion Along a Straight Line) a! "otion "otion with constant constant velocit velocity y #uniform #uniform motion! motion! s =ϑ t  b! "otion "otion with with constant constant accelerati acceleration on ϑ f =ϑ i+ at  1

2

s =ϑ i t + a t  2

2

ϑ f 

2

=ϑ i + 2

as

c! $ree ree $all $allin in %ody %ody ϑ =¿ 1

2

h = g t  2

2

ϑ 

=2

gh

Note: From From motio motion n with with consta constant nt accele accelera ratio tion, n, set: set: s = h , and

a =g  to get the free-fall formulas.

d! "otion "otion with with variable variable accele acceleratio ration n dϑ  a= dt  ϑ =

ds dt 

ads = vdv

&here s ' distance

ϑ i= 0

,

ϑ f =ϑ 

,

h ' heiht ϑ ' velocity ϑi ' initial velocity ϑf  ' (nal velocity a ' acceleration g ' acceleration due to ravity # g ' ).*+ ms- in I ' /-.- fts - in 0nlish! t  ' time 1ote2 3 a is positive #4! if ϑ is increasin #accelerate!. 3 a is neative #5! if ϑ is decreasin #decelerate!. 3 a is positive #4! if the particle is movin downward. 3 a is neative #5!if the particle is movin upward. 6seful conversion factors2 From 7ilometers per hour #8mhr or 8ph! "eters per second #ms or mps! "iles per hour #mihr or mph! $eet per second #fts fps!

To "eters per second #ms or mps! 7ilometers per hour #8mhr or 8ph! $eet per second #fts fps! "iles per hour #mihr or mph!

Multipl B !"#.$ #.$ %%"!& !&"%%

'ro(lems) #Problem et + Ta8e 9ome :uiz +! 3

4

+. The motion of a particle is iven by

s =2 t  −

t 

6

−2

2

t  , where s is in feet

and t is in seconds. ;ompute the values of linear velocity and acceleration after - seconds. -. A stone is dropped down a well and < sec later, the sounds of the splash is heard. If the velocity of sound is ++-= fts #/>+./?@ ms!, what is the depth of the well #Ans. / m!

/. A hot5air balloon is ascendin at the rate of += 8ph. If a bomb is dropped from the balloon such that it ta8es * seconds to reach the round, determine the balloonCs altitude when the bomb was released. #Ans. -)+.@ m! >. Dn a certain stretch of trac8, trains run at @= mph #)@.<@ 8ph!. 9ow far bac8 of a stopped train should be a warnin torpedo be placed to sinal an oncomin train Assume that the bra8es are applied at once and retard the train at the uniform rate of - fts - #=.@+ ms-!. #Ans. +)/@ ftB <*).? m! <. A stone is thrown vertically upward and return to earth in += sec. &hat was its initial velocity and how hih did it o #Ans. >).=< ms, +--.@-< mB +@+ fts, >=-.< ft! @. A ball is dropped from the top of a tower *= ft #->./* m! hih at the same instant that a second ball is thrown upward from the round with an initial velocity of >= fts #+-.+) ms!. &hen do they pass &here do they pass based on the top of the tower #Ans. They pass each other after - seconds at @>.> ft from the top of  the tower.! #Ans. They pass each other after - seconds at +).@- m from the top of the tower.! ?. A stone is dropped from a captive balloon at an elevation of +=== ft #/=>.* m!. Two seconds later another stone is thrown vertically upward from the round with a velocity of ->* fts #?<.@ ms!. If  ' /- fts- #).?< ms-!, when and where the stones pass each other #Ans. The stones will pass each other < seconds after the (rst stone was dropped from the captive balloon E the stones will meet at a point @== ft above the roundB The stones will pass each other < seconds after the (rst stone was dropped from the captive balloon E the stones will meet at a point +*-.)-< m above the round.! *. A stone is thrown vertically upward from the round with a velocity of >*./ fts #+>.?- ms!. Dne second later another stone is thrown vertically upward with a velocity of )@.@ fts #-).>> ms!. 9ow far above the round will the stones be at the same level #Ans. /@.=@> ftB +=.))> m! ). A ball is shot vertically into the air at a velocity of +)/.- fts #<*.) ms!. After > seconds, another ball is shot vertically into the air. &hat initial velocity must the second ball have in order to meet the (rst ball /*@.> ft #++?.* m! from the round #Ans. +<*.@? ftsB >*./@ ms!