CHAPTER 4 Motion in Two Dimensions 1. PROJECTILE MOTION 2. CALCULATION OF TIME OF FLIGHT 3. CALCULATION OF RANGE & MAX. RANGE. 4. CALCULTION OF ALTITUDE 5. CALCULATION OF TRAJECTORY 6. CIRCULAR MOTION 7. ANFGULAR DIPLACEMENT 8. ANGULAR !ELOCITY 9. ANGULAR ACCELERATION 10.RELATION "ET#EEN LINEAR AND ANGULAR !ELOCITIE 11.CENTRIPETAL ACCELERATION 12.CENTRIPETAL FOCE 13.E$UATION 14.DIMENION. 15.HORT $UETION AND AN#ER
DESCRIPTIVE PART Projectile motio!
An o%e't is t()own *t 'e)t*in *n+,e wit( +)o-n ,e/e,0 t(e motion o1 *n o%e't is -ne) t(e in1,-en'e o1 +)*/it2. -'( -'( t23e o1 motion is '*,,e P)oe'ti,e motion5. T(e t()own o%e't is '*,,e t(e 3)oe'ti,e *n its 3*t( is '*,,e t(e t)*e'to)25. E"#l$$tio! Let -s 'onsie) * 3)oe'ti,e6 w(i'( is t()own *t *n *n+,e θ wit( t(e (o)i7ont*,6 (*/e initi*, t*8e o11 /e,o'it2 / o5. A 3)oe'ti,e '*))ies '*))ies o-t two motions9 :. A 'onst* 'onst*nt nt (o)i (o)i7on 7ont*, t*, /e,o'i /e,o'it2 t2 /o; < /o Cos θ6 w(i'( )em*ins 'onst*nt6 %2 ne+,e'tin+ *i) 1)i'tions6 =. *n * /e)ti'* /e)ti'*,, *''e,e)* *''e,e)*tion tion -ne) -ne) t(e t(e in1,-en'e in1,-en'e o1 +)*/it2 +)*/it2 +5. +5. -'( t23e o1 two imension*, motions is '*,,e 3)oe'ti,e motion5. ome e;*m3,es o1 3)oe'ti,e motion *)e9 A %-,,et 1i)in+ 1)om +-n6 • Lon+ -m3in+6 • • "om% )o33in+ 1)om *i)3,*ne.
• C$lc%l$tio o& Time o& &li'(t! Tot*, Tot*, time t*8en %2 t(e 3)oe'ti,e to 'o/e) t(e t)*e'to)2 is '*,,e Time o1 1,i+(t56 *n enote %2 T5.
/ 1 < /i > + t / Y < /oY > ?@+ t #(e)e6
/ Y < / 1 < B
*t 3e*8 3oint *n
/ oY < / i < / o in
6 -st *1te) ,e*/in+ t(e +)o-n.
B < / o in in > ?@+ ?@+ t
T(e)e1o)e6 C$lc%l$tio o& m$"im%m r$'e!
T(e m*;im-m (o)i7ont*, (o)i7ont*, ist*n'e 'o/e)e %2 t(e 3)oe'ti,e is '*,,e M*;im-m M*;im-m R*n+e56 enote %2 R R m*;. m*;.5. I1 t(e /*,-e o1 in = θ is m*;im-m6 t(en )*n+e wi,, %e % e m*;im-m. It me$)*
in = < : + t < /o sin in t<
O)
/o
[ sin ]
+
6 T(is is * time 1o) -3 w*) motion6 to )e*'( *t t(e m*;im-m (ei+(t. An t(e
t< time )e-i)e in own w*) motion6 to )e*'( *t t(e s*me ,e/e, on +)o-n is6 Hen'e6 T< t>t /o
T< + T<=
we +et6 C$lc%l$tio o& R$'e!
[ sin ]
/o +
/o
> +
[ sin ]
[ sin ] 6 It is t(e tot*, time o1 1,i+(t.
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
/o +
[ sin ] .
Tot*, Tot*, (o)i7ont*, ist*n'e 'o/e)e %2 t(e 3)oe'ti,e to (it t(e t*)+et6 is '*,,e R*n+e56 enote %2 R5. T(e (o)i7ont*, /e,o'it2 'om3onent o1 3)oe'ti,e motion 'ontin-es -n'(*n+e ne+,e'tin+ *i) 1)i'tion. Hen'e t(e (o)i7ont*, ist*n'e o1 1,i+(t is -ne) 'onst*nt (o)i7ont*, /e,o'it2 / o;6 +i/en %26
< / o; T
6 1o) -ni1o)m (o)i7ont*, motion. R < / o Cos Cos T @ @ @ @ @ @ @ @ @ e. ?* :
( < /i t >
B < / o2 t >
: = :
=
=
: =
+ t=
( @ +) t=
+ t = < / o sin t
+ t< t< /osin sin
/ t < = o sin + /o
T< =
[ in ]
+
O)
*s tot*, time o1 1,i+(t. / R < / o Cos = o in in + "2 3-ttin+ t(e /*,-e o1 T5 in e-*tion ?*6 we +et6 =
R< R<
/o
[ = in
+ /o=
#e +et6
Cos Cos ]
[ in = ]
+
6 -se 1o) '*,'-,*tion o1 R*n+e. @:
= < in ?: ?: = < GBo
<
GBo
< 4H
= o
o R < R m*; T(is s(ows t(*t6 i1 < 4H 6 t(en It is t(e m*;im-m )*n+e5. /=o in GBo R m*; < +
R m*; <
/ =o +
[ :]
=
R m*; <
/o +
C$lc%l$tio o& $ltit%+e!
T(e m*;im-m /e)ti'*, ist*n'e 'o/e)e %2 t(e 3)oe'ti,e is '*,,e A,tit-e5. T(e own w*) *''e,e)*tion o1 t(e 3)oe'ti,e is t(e s*me *s 1)ee 1*,,in+ %o2 *n t*8es 3,*'e ine3enent o1 (o)i7ont*,
/
motion. T(e P)oe'ti,e (*s initi*, t*8e o11 /e)ti'*, /e,o'it2 oY *n 1in*, /e)ti'*, /e,o'it2 *t 3e*8 3oint %e'omes 7e)o *2 6/Y < /1 ? @ + tI = #e 8now t(*t6 : ( < /oY t @ + tI = @ @ @ @ @ @ @ @ @ e. ?i #(e)e6
/ 1 < /i > + t
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
/ Y < / oY > ? @ + t B < / o in in > ? @+ t
O)
+ t < /o in in t<
/o +
[ in ]
Hen'e6 "2 3-ttin+ t(e /*,-e o1 t5 in e-*tion ?i6 we +et6 / : /o ( < /o in o in @ + in + = +
(< (< (<
/ o= + /
= o
+
inI ] @ [ inI
[ inI ]
@
: =
/ =o
+
+I
=
inI
/ inI inI = + = o
:
: /o=
in inI = +
C$lc%l$tio o& Tr$jector,! A 3)oe'ti,e 'o/e)s * (o)i7ont*, ist*n'e *,on+ wit( /e)ti'*, is '*,,e T)*e'to)256 me*ns 3*)*%o,i' '-)/e o1 3)oe'ti,e. ( < /i t >
#e 8now t(*t6
: =
+ tI
Y < / o i in
O)
:
(T) @
=
+
( TI )
@ @ @ @ @ @ @ @ @ e. ?:
X / o Cos
"2 3-ttin+ t(e /*,-e o1 o1 T5 in e-*tion e-*tion ?:
we +et6 +et6
X Y < /o in in Cos /o Co
@
in Y< X Cos Co
Y < X [* ] @
: =
XI
[ T*n ]
@
: =
=
= / Cos : + @ X= = = = / o Cos
in Y< X Cos Co
Y
: X @ + = /o Co Cos
:
+
X=
= o
=
CosI / Co
XI
+
= o
[ %]
6 T(is e-*tion is '*,,e '*,,e P*)*%o,* P*)*%o,* e-*tion5. + % < = * < [ T*n ] / o CosI "e'*-se6 & -i&orm circ%l$r motio! 1. !e,o'it2 !e,o'it2 )em*ins 'onst*nt in m*+nit-e %-t /*)ies in i)e'tion 2. T(e *''e,e)*tion is *,w*2s no)m*, to t(e /e,o'it2 /e'to). 3. T(e *''e,e)*tion is *,w*2s i)e'te tow*)s t(e 'ent)e o1 t(e 'i)'-,*) 3*t(. Ci)'-,*) motion is '*-se w(en *n o%e't e;3e)ien'es -st t(e )i+(t *mo-nt o1 net 1o)'e i)e'te tow*) t(e 'ente) o1 t(e 'i)',e o) '-)/e. T(e *''e,e)*tion *''e,e)*tion o1 *n o%e't in -ni1o)m 'i)'-,*) 'i)'-,*) motion is *,w*2s tow*) t(e 'ente) o1 t(e 'i)',e. 'i)',e. T(e *''e,e)*tion *n t(e 1o)'e '*-sin+ it *)e '*,,e 'ent)i3et*, w(i'( me*ns 'ente)@see8in+. 'ente)@see8in+. Cent)i3et*, 1o)'e '*n %e 3)o/ie to * 3,*net %2 +)*/it26 to * '*) %2 1)i'tion %etween w(ee,s *n t(e )o*6 o) to * %*,, t(e t(e tension on * st)in+. in'e t(e i)e'tion o1 t(e /e,o'it2 o1 t(e o%e't is 'ontin-o-s,2 '(*n+in+ tow*) t(e 'ente) o1 t(e 'i)',e6 t(e o%e't is s*i to e;3e)ien'e 'ent)i3et*, *''e,e)*tion *''e,e)*tion * e/en t(o-+( its s3ee is 'onst*nt.
ome e;*m3,es o1 'i)'-,*) motion9 A motion o1 t(e e*)t( *n ot(e) 3,*nets *)o-n s-n6 Motion o1 moon *)o-n e*)t(6 motion o1 e,e't)on *)o-n n-',e-s6
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
TEC/ECA TER REATIVE DEI/ITI/S 1.
Rot$tio$l mo motio! A )i+i o%e't (*s 'i)'-,*) motion *%o-t * 1i;e )e1e)en'e 3oint 3*ss t()o-+( t(e )i+i o%e't. T(e motion o1 t(e )i+i o%e't *s * w(o,e is '*,,e Rot*tion*, motion5.
2.
A'%l$r motio! An o%e't mo/es in * 'i)',e *%o-t 1i;e )e1e)en'e 3oint .T(e *n+,e m*e %2 t(e *)' *n t(e )*i-s. -'( * motion is '*,,e An+-,*) motion5.
3.
A'%l$r +i)#l$cemet)! An *n+,e s-%tene %etween *n *)' *n )*i-s o1 * 'i)',e is '*,,e An+-,*) is3,*'ement56 enote %2 θ5. T(is -nit is '*,,e )*i*n me*s-)e. It is imension,ess. T(e *n+-,*) is3,*'ement is3,*'ement o1 *n o%e't is t(e6 )*tio %etween *)' ,en+t( ,en+t( *n )*i-s o1 t(e 'i)',e w(en *n o%e't o%e't * mo/es 1)om 3osition 3osition A to 3osition " in s(o)t time. *)' ,en+t( A" < )*.i-s <
4.
s )
Coer)io o& +e're 'ree ito r$+i$)! T(e *n+,e s-%tene %2 *n *)' o1 * 'i)',e w(ose w(ose *)' ,en+t( is e-*, to t(e )*i-s o1 s*me 'i)',e is '*,,e one )*i*n5. T(e tot*, 'i)'-m1e)en'e is ?= π) is s-%tene %2 K Bo )*.i*n *nss ( = ) ) )*.i JKBo < ) Hen'e6 KBo < = π )*i*ns JKBo =
< :N )*.i*n
. o < : )*i*n :o < B.B:4 )*i*ns An6 one )e/o,-tion < =π 5.
A'%l$r elocit,! T(e */e)*+e *n+-,*) /e,o'it2 o1 * )ot*tin+ o%e't is e1ine to %e5 t(e *n+-,*) ist*n'e i/ie %2 t(e time t*8en to t-)n t()o-+( t(is *n+,e5 enote %2 ω56 its -nit is )*i*ns se'.
Let -s 'onsie) *n o%e't mo/in+ in * 'i)',e wit( -ni1o)m s3ee *%o-t 1i;e )*i-s )5. T(e *n+-,*) is3,*'ement θ is 1)om 3osition P to $
is in time t5. T(e )*te o1 '(*n+e o1 o1 *n+-,*) is3,*'ement is3,*'ement is t(e *n+-,*) /e,o'it2. /e,o'it2. Its imension is T@:. Q<
in'e6 Q<
= @ : t = @ t:
t It is /e'to) -*ntit20 i)e'tion is ete)mine %2 )i+(t (*n )-,e. I1 2o- '-), t(e 1in+e)s o1 2o-) )i+(t (*n to 1o,,ow t(e i)e'tion o1 t(e )ot*tion6 t(en t(e i)e'tion o1 t(e *n+-,*) /e,o'it2 /e'to) is ini'*te %2 2o-) )i+(t t(-m%
6.
I)t$t$eo%) $'%l$r elocit,! #(en time *33)o*'(es to 7e)o6 t(e *n+-,*) /e,o'it2 *t /e)2 s(o)t time inte)/*, is '*,,e Inst*nt*neo-s *n+-,*) /e,o'it25 /e,o'it25
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
Q < ,im ,im ∆t →B t 7.
•
A'%l$r $c $cceler$tio! T(e '(*n+e o1 *n+-,*) *n+-,*) /e,o'it2 in -nit time is '*,,e *n+-,*) *n+-,*) *''e,e)*tion5. *''e,e)*tion5. It is enote enote %2 α5. Its -nit is )*i*nsSse'.
Let -s 'onsie) *n o%e't is mo/in+ in * 'i)',e6 wit( *n+-,*) /e,o'it2 ω: in time t:6 *1te) ∆t se'ons t(e *n+-,*) /e,o'it2 %e'omes ω= in time t=. T(e '(*n+e o1 *n+-,*) /e,o'it2 in -nit time is An+-,*) *''e,e)*tion5. *''e,e)*tion5. Its imension is T @ . <
<
Q= @ Q: t = @ t:
Q t
8. I)t$ I)t$t t$e $eo%) o%) $'%l $'%l$r $r $ccel $cceler$ er$tio tio! ! #(en time *33)o*'(es to 7e)o6 t(e *n+-,*) *''e,e)*tion *t /e)2 s(o)t time in te)/*, is '*,,e Inst*nt*neo-s *n+-,*) *''e,e)*tion5. ∆Q ∆ < ,im ,im ∆ →B ∆t t
I1 t(e *n+-,*) *''e,e)*tion is -ni1o)m6 we (*/e t(*t t(e */e)*+e *''e,e)*tion *''e,e)*tion is t(e s*me *s t(e inst*nt*neo-s *n+-,*) *''e,e)*tion. T(e *n+-,*) *''e,e)*tion *n *n+-,*) /e,o'it2 *)e /e'to) -*ntities t(ei) i)e'tion is ete)mine %2 Ri+(t (*n )-,e5. A''o)in+ to t(is )-,e6 “Place fingers of right hand on to wheel, by curling the fingers in the direction direction of rotation. The thumb point perpendicular to the wheel indicates the direction of angular quantities”. T(e ',o'8@wise )ot*tion is i)e'te in to t(e 3*3e)6 *n *nti@ wise )ot*tion is i)e'te o-t o1 3*3e). T(e i)e'tion o1 *n+-,*) /e,o'it2 *n *n+-,*) *''e,e)*tion *''e,e)*tion is s*me in i)e'tion6 *n+-,*) /e,o'it2 'ontin-o-s,2 in')e*ses in')e*ses in time6 w(i'( is 3ositi/e *n+-,*) *''e,e)*tion ?> α *n 1o) t(e o33osite i)e'tions t(e *n+-,*) /e,o'it2 'ontin-o-s,2 e')e*ses in time6 w(i'( is ne+*ti/e *n+-,*) *''e,e)*tion ?@ α. 9.
Rel$ Rel$ti tio o et etee ee lie lie$r $r $+ $+ $'% $'%l$ l$rr elo eloci citi tie) e)!! T(e *n+-,*) s3ee o1 t(e )ot*tin+ o%e't is to %e )e,*te wit( t(e t*n+enti*, s3ee o1 * 3osition 3oint A o1 *n o%e't "e'*-se6 3oint A mo/es mo/es in * 'i)',e6 t(e ,ine*) ,ine*) /e,o'it2 /9 is *,w*2s *,w*2s t*n+ent to t(e 'i)'-,*) 'i)'-,*) 3*t( *n (en'e (en'e is '*,,e t*n+enti*, /e,o'it2. /e,o'it 2. A1te) A1t e) ∆t5 se'ons t(e *n+-,*) is3,*'ement is ∆θ. <
#e 8now t(*t6
s )
: s m < ,im ∆,i→ B ∆t B ∆ → ) ∆t "2 i/iin+ ∆t5 %ot( sies we +et6 t
t
Q<
:
[ /]
)
ω )
/ <)Q 10.
T$'eti$l e elocit,! T(e 3)o-'t %etween )*i-s o1 * 'i)',e *n *n+-,*) /e,o'it2 is '*,,e T*n+enti*, T*n+enti*, /e,o'it250 enote %2 /t5. T(e t*n+enti*, /e,o'it2 o1 * 3oint on * )ot*tin+ )i+i o%e't e-*,s t(e 3e)3eni'-,*) ist*n'e o1 t(*t 3oint 1)om t(e *;is o1 )ot*tion m-,ti3,ie %2 t(e *n+-,*) /e,o'it2. T(e)e1o)e6 T(e)e1o)e6 *,t(o-+( e/e)2 3oint on t(e )i+i o%e't (*s t(e s*me *n+-,*) /e,o'it26 not e/e)2 3oint (*s t(e s*me ,ine*) s3ee %e'*-se )5 is not t(e s*me 1o) *,, 3oints on t(e o%e't. T(e i)e'tion is t*n+ent to its 'i)'-,*) 3*t(. Its m*+nit-e is +i/en %26
/t < ) Q
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
11.
T$'eti$l $cceler$tio! T(e t*n+enti*, *''e,e)*tion o1 t(e 3oint P '*n )e,*te to t(e *n+-,*) *''e,e)*tion o1 t(e )ot*tin+ )i+i o%e't. T(e 3)o-'t %etween )*i-s o1 * 'i)',e *n *n+-,*) *n+-,*) *''e,e)*tion *''e,e)*tion is '*,,e T*n+enti*, T*n+enti*, *''e,e)*tion50 *''e,e)*tion50 enote %2 * t5. -33ose *n o%e't is mo/in+ *%o-t 1i;e *;es6 it '(*n+es its *n+-,*) /e,o'it2 in ∆t5 se'ons.
/t < ) Q
#e 8now t(*t
/ Q ,im t < ) ,im ∆ →B ∆ →B ∆t ∆t "2 i/iin+ ∆t %ot( sies6 #e +et6 t
t
* <)
t Hen'e 12. Time #erio+! • T(e time )e-i)e to 'om3,ete one )ot*tion is '*,,e Time 3e)io56 enote %2 T5. #(e)e s
DESCRIPTIVE PART 13.
Cetri#et$l $cceler$tio! $cceler$tio! Cetri#et$l T(e *''e,e)*tion 3e)3eni'-,*) on to ,ine*) /e,o'it26 i)e'te tow*)s t(e 'ent)e6 o1 'i)'-,*) 3*t( is '*,,e Cent)i3et*, *''e,e)*tion5. E"#l$$tio! Let -s 'onsie) * stone is *tt*'(e *t one en o1 * st)in+ *n w(i),e )o-n *t 'onst*nt s3ee in * 'i)'-,*) 3*t( t(e i)e'tion o1 its /e,o'it2 'ontin-o-s,2 '(*n+es. T(e)e1o)e6 it is s*i t(*t * %o2 mo/in+ on * 'i)'-,*) 3*t( (*s *''e,e)*tion. $t(em$tic$l +eri$tio!
-33ose * stone is to %e w(i),e in * 'i)',e o1 )*i-s R5 *t 'onst*nt *n+-,*) s3ee ω5. Its ,ine*) t*n+enti*, /e,o'it2 / 5 '(*n+es6 in time ∆t se'ons. #(en6 * stone '(*n+e 3osition 1)om A to 3osition "6 t(*n t(e /e,o'ities *)e /A *n / " *t t(e )es3e'ti/e 3osition 3oints. T(e m*+nit-es o1 /e,o'ities *)e s*me6
/
<
/A
<
/"
As t(e O5 is t(e o)i+in o1 * 'i)',e6 )*w t(e )e3)esent*ti/e ,ines OA 5 1o) / A 5 *n O" 5 1o) / " 56 *''o)in+ initi*, to te)min*, )-,e6 / A > Hen'e6
∆/ ∆/
< / " 6 is t(e )es-,t*nt /e,o'it2. T(e T(e '(*n+e in /e,o'it2 is
∆/ %etween two +i/en +i/en 3ositions.
< / " / A
T(e /e)2 sm*,, *n+-,*) is3,*'ement %etween / A *n / " is ∆θ in /e)2 s(o)t time ∆t. / < / T(e)e1o)e6 Di/ie %ot( sies %2 ∆t0 it is time inte)/*,6 in w(i'( /e,o'it2 *n *n+-,*) is3,*'ement '(*n+e6 : / < ,im ,tim → B t / t →B t #e +et
[ Q]
:
[ *] / : / ) < / [ * ] / = ) < [ * ] <
T(e *''e,e)*tion is tow*)s t(e 'ent)e6 8nown *s 'ent)i3et*, *''e,e)*tion. *' <
/
=
) Hen'e6 T(is is m*+nit-e o1 'ent)i3et*, *''e,e)*tion * ' *n *,w*2s i)e'te tow*) t(e 'ent)e o1 'i)',e.
*' <
( ) Q)
*' < ) Q
)
=
"e'*-se6 / < ) ω
=
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
= *' < ) T
=
=
Q<
T
"e'*-se6
4 = ) = T
*' <
=
1<
=
*' < 4 ) 1
T(e 'ent)i3et*, *''e,e)*tion T(-s6
is *,so '*,,e 3e)3eni'-,*) *''e,e)*tion5 *''e,e)*tion5 ? = = 4 ) / = = = *' < < ) Q < = = 4 ) 1 ) T
*<
14.
T
*'
An t*n+enti*, *''e,e)*tion5 *''e,e)*tion5 %e6
An
"e'*-se6
:
*t
5 is t(e 3*)*,,e, *''e,e)*tion
*P
*⊥
5. T(e m*+nit-e *n i)e'tion o1 )es-,t*nt *''e,e)*tion wi,,
* '= > * =t
* < t*n @: t * '
Cetri#et$l
&orce! A 1o)'e 3-,,in+ *n o%e't tow*) t(e 'ente) o1 * 'i)'-,*) 3*t( *s t(e o%e't mo/es *)o-n t(e 'i)',e is '*,,e * 'ent)i3et*, 1o)'e. An o%e't '*n t)*/e, in * 'i)',e on,2. E"#l$$tio! Let -s 'onsie) * %o2 o1 m*ss m56 is *tt*'(e *t one en st)in+ w(i'( is w(i),e in * 'i)',e6 t(e tension in t(e st)in+ 3)o/ies * 'ent)i3et*, 'ent)i3et*, 1o)'e6 e;e)te e;e)te %2 t(e st)in+ is -n%*,*n'e -n%*,*n'e 1o)'e. A %o2 is mo/in+ *,on+ * 'i)'-,*) 'i)'-,*) 3*t( wit( -ni1o)m -ni1o)m s3ee. T(e i)e'tion o1 /e,o'it2 *t *n2 3oint on t(e 'i)',e is *,on+ t(e t*n+ent t*n+ent to t(e 'i)',e. T(-s t(e i)e'tions o1 motion o1 t(e %o2 '(*n+e 1)om 3oint to 3oint. T(e 'ent)i3et*, 1o)'e (*s no 'om3onent *'tin+ on t(e %o2 in t(e i)e'tion o1 motion. T(is me*ns t(*t t(e i)e'tion o1 'ent)i3et*, 1o)'e *,w*2s 3e)3eni'-,*) to t(e i)e'tion o1 motion o1 t(e %o2 *n i)e'te tow*)s 'ent)e o1 'i)',e. "2 se'on ,*w o1 motion6 F' < m *' in'e6 -n%*,*n'e 1o)'e is not 7e)o0 t(e %*,, m-st (*/e 'ent)i3et*, *''e,e)*tion.
Hen'e6 O)
F' < m
/= )
F' < m ) Q=
4 = ) = T
F' < m
⇒
F' < 4 = m ) 1 =
F' < m
/= )
4 = ) = 4 = m) 1 = = T
or e"$m#le! E,e't)on )e/o,/es )o-n t(e n-',e-s. T(e 1o)'e o1 *tt)*'tion %etween t(e n-',e-s *n t(e e,e't)on 3)o/ies t(e )e-i)e 'ent)i3et*, 1o)'e. T(e 3,*nets mo/e *)o-n t(e s-n. T(e 1o)'e o1 *tt)*'tion %etween t(e s-n *n t(e 3,*net 3)o/ies t(e ne'ess*)2 'ent)i3et*, 1o)'e. T(e moon )e/o,/es )o-n t(e e*)t(. T(e 1o)'e o1 *tt)*'tion %etween t(e e*)t( *n t(e moon 3)o/ies t(e )e-i)e 'ent)i3et*, 1o)'e. A stone *t one en o1 st)in+ is w(i),e6 * 'ontin-o-s 'ent)i3et*, 1o)'e e;e)te inw*) %2 me*ns o1 t(e st)in+. 15. 16.
Cetri&%'$l &orce! T(e o-tw*) )e*'tion *'tin+ on -s is '*,,e t(e Cent)i1-+*, 1o)'e56 w(i'( tens to mo/e t(e stone *w*2 1)om t(e 'i)'-,*) 3*t(. Consie) * stone w(i),e *t t(e en o1 st)in+. "2 me*ns o1 t(e st)in+ we 'omm-ni'*te t(e 'ent)i3et*, 'ent)i3et*, 1o)'e to t(e stone. A''o)in+6 A''o)in+6 t(i) ,*w o1 motion t(e )e*'tion o1 t(e stone *'ts o-tw*) i)e'tion. T(is s(ows t(*t t(e ne'ess*)2 'ent)i3et*, 1o)'e to t(e stone t()o-+( t(e st)in+ w(i,e *n e-*, *n o33osite )e*'tion o1 t(e stone *'ts on -s i)e'te *w*2 1)om t(e 'ent)e 'ent)e o1 'i)',e 'i)',e ? F ' < @ F + . T(is )e*'tion 1o)'e is t(e 'ent)i1-+*, 1o)'e5. T(e 'ent)i1-+*, 1o)'e *'ts *s ,on+ *s t(e 'ent)i3et*, 1o)'e is 3)esent.
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
E%$tio) o& circ%l$r motio! F+ < @ m
/= )
⇒
F+ < @ m ) Q =
⇒
4 = ) F+ < @ m = T
⇒
F+ < @ 4 = m ) 1 =
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DIE/SI/ PSICA -A/TIT
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DIE/SI/
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( LT@: ) = @= < L T < L [ ] @= @=
R<
+
R$'e ;R<
LT
LT@: < [ T ] LT@=
/o
t<
+
Time
: /o=
(<
= +
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=
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( LT ) = @= < L T < L [ ] LT @= LT @=
mete)
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=
sin
mete)
s A'%l$r +i)#l$cemet
< ) Q <
A'%l$r elocit,
< A'%l$r $cceler$tio
t Q t
Dimension,ess no .imention < T @: [ T] T < T @= T [ ]
R*i*n o) e+)ee 3e) se'
@:
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
3e) se'@=
TI ? 1 > i =
Q < T$'eti$l elocit, T$'eti$l $cceler$tio
Cetri#et$l $cceler$tio
Cetri#et$l &orce
t
F' <
/
@: LT LT
[ L] T <
LT LT
@=
*t < ) *' <
[ L] T@: <
m 3e) se'
@=
m 3e) se'@=
( LT@: ) = = @= < L T < LT @= [ L] [ L]
=
) m/ =
[ M] ( LT@: )
)
[ L]
=
< ML T < MLT @= [ L] =
m 3e) se'@=
@=
Newton
S(ort %e)tio) $+ A)er) . =1! A omer +ro#) it) om (e* it i) ertic$ll, $oe it) t$r'et* it mi))e) it. E"#l$i t(e )t$temet. A %om% 1o,,owe t)*e'to)26 *1te) *1te) )o33in+ 1)om t(e %om%e). o t(*t it '*nVt (it t(e t*)+et. o t(*t * %om%e) )o3s its %om% w(en6 it is /e)ti'*,,2 *%o/e its t*)+et6 it misses it. . =2! I ro$+ j%m#i' +oe) it m$tter (o ,o% j%m#> ?($t &$ctor) +etermie t(e )#$ o& t(e j%m#> In %)o* -m3in+ it oesnVt m*tte) (ow (i+( we -m3. T(e 1*'to)s o1 s3*n o1 t(e -m3 ete)mine t(e initi*,. !e,o'it2 !e,o'it2 -st *1te) ,e*/in+ t(e +)o-n6 *n+,e o1 s3*n o1 -m3 -m3 *n *''e,e)*tion -e to +)*/it2. . =3! De&ie t(e time &li'(t $+ rite it) e"#re))io> e"#re))io> / T < = o in +
Tot*, Tot*, time o1 3)oe'ti,e to 1o,,ow t)*e'to)2 is '*,,e time o1 1,i+(t5. . =4! ?(, $ omer +oe) ot +ro# t(e om) (e* it i) ertic$ll, $oe t(e t$r'et> "e'*-se *1te) )o33in+ t(e %om% 1)om t(e %om%e) * %om% 1o,,owe t)*e'to)2. o t(*t * %om%e) oes not )o3 t(e %om%s w(en6 it is /e)ti'*,,2 *%o/e t(e t*)+et. . =5! ?(e $ c$r t$@e) $ t%r* t(e #$))e'er) e"#eriece) &orce $cti' o t(em)ele) $$, &rom o& c%re* (,> "e'*-se6 3*ssen+e)sV in mo/in+ '*) t)2 to %*,*n'e t(emse,/es w(en * '*) t*8es * t-)n. o t(*t6 w(en * mo/in+ '*) t-)ns )o-n * 'o)ne) to t(e ,e1t t(e 'ent)i3et*, 1o)'e *33,ie %2 t(e '*) *n6 t(e 3*ssen+e)Vs in t(e o33osite i)e'tion 3)o/ies t(e 'ent)i1-+*, 1o)'e. . = 6! I ($t +irectio +oe) t(e m%+ &l, o&& t(e tire) o& moi' c$r $+ (,> In t(is '*se )ot*t*)2 motion o1 t(e w(ee, 3)o/ies t(e 'ent)i1-+*, 1o)'e *n t(e 1o)'e %etween m- *n w(ee, 3)o/ies 'ent)i3et*, 1o)'e. As t(e s3ee o1 w(ee, in')e*ses *t * 'e)t*in s3ee t(e )e*'tion 1o)'e e;'ees t(e 'ent)i3et*, 1o)'e .T(e m- wi,, 1,2 o11 t(e t*n+enti*, i)e'tion to t(e w(ee,. . =7! Doe) t(e (oriBot$l elocit, com#oet o& #rojectile #rojectile motio rem$i co)t$t* (,> Yes6 Yes6 t(e (o)i7ont*, /e,o'it2 'om3onent o1 3)oe'ti,e motion )em*ins 'onst*nt6 % 2 ne+,e'tin+ *i) 1)i'tion. . = 8! A o+, i) moi' $lo' $ circle it( co)t$t )#ee+ i< i) it moi' it( %i&orm elocit,> ii< I) it moi' it( co)t$t $cceler$tio* ($t) +irectio o& $cceler$tio> A %o2 is mo/in+ *,on+ * 'i)',e wit( 'onst*nt s3ee i t(e i)e'tion o1 /e,o'it2 '(*n+e *t e/e)2 3oint *n t(e m*+nit-e )em*ins s*me. o t(*t * %o2 (*s no -ni1o)m /e,o'it2 .ii A %o2 (*/e 'onst*nt *''e,e)*tion *''e,e)*tion t(*t is 'ent)i3et*,. T(e i)e'tion is tow*)s t(e 'ent)e o1 'i)',e. . = 9! Proe t($t* r / < ) ω o) / < ) t
[ = ] )*i*ns / < ) t . /<
we +et6
ist*n'e t
Hen'e s(own
. =10! E"#l$i t(e )t$temet or@ +oe , cetri#et$l &orce i) Bero.
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
T(e 'ent)i3et*, 1o)'e is *,w*2s i)e'te to t(e 'ent)e o1 * 'i)',e6 w(i'( is 3e)3eni'-,*) o to 'i)'-m1e)en'e. 'i)'-m1e)en'e. #o)8 #o)8 one is t(e 3)o-'t %etween 1o)'e *n is3,*'ement. #o)8 < F' 'os θ #o)8 < F' ? = π ) Cos Bo #o)8 #o)8 < B6 so t(*t no wo)8 wi,, %e one -e to 'ent)i3et*, 1o)'e. . =11! A'%l$r elocit, i) $ ector %$tit,. ?($t i) it) +irectio> T(e i)e'tion o1 *n+-,*) /e,o'it2 is ete)mine %2 )i+(t (*n )-,e6 *''o)in+ to t(is )-,e6 3,*'e t(e 1in+e)s o1 )i+(t (*n in t(e i)e'tion o1 )ot*tion o1 is' .t(e t(-m% ini'*tes t(e i)e'tion o1 *n+-,*) /e,o'it2. . =12! S(o t($t $'le o& oe r$+i$ i) e%$l to 57.295o>
θ
<
'i)'-,*) 3*t( )
( = ) )
)*.i*ns
) KB o < o KB < = π )*i*ns
o)
JKBo = < : )*i*n
o)
JKBo = ? J.:4 < : )*i*n
o) Hen'e6
. o < : )*i*n
. =13! ?($t i) t$'eti$l elocit,> T(e ,ine*) /e,o'it2 *t *n2 3oint on t(e 'i)'-m1e)en'e o1 * 'i)',e6 '*,,e t*n+enti*, /e,o'it2. It is 3)o-'e %etween )*i-s o1 * 'i)',e *n *n+-,*) /e,o'it26 s*2 / t < ) ω . =14! Cetri#et$l &orce $+ cetri&%'$l re$ctio $re e%$l i m$'it%+e %t o##o)ite i +irectio. Do $l$ce e$c( ot(er> T(e 'ent)i3et*, *n 'ent)i1-+*, 1o)'es *)e e-*, in m*+nit-e %-t o33osite in i)e'tion *,on+ s*me ,ine o1 *'tion. A''o)in+ A''o)in+ to t(e t(i) ,*w o1 motion6 t(ese 1o)'es *)e *'t *s t(e *'tion *n )e*'tion. . =15! I) it ($r+er &or $ c$r to t%r ro%+ $ corer $t (i'( )#ee+ t($ $t lo )#ee+> ?(,. ?(,. #(en * '*) t-)ns )o-n * 'o)ne) t(e 1o)'e o1 1)i'tion %etween t(e w(ee,s o1 '*) *n t(e )o* 3)o/ies ne'ess*)2 'ent)i3et*, 1o)'e. As t(e s3ee in')e*ses t(e 'ent)i3et*, 1o)'e in')e*ses *n *t * 'e)t*in s3ee t(e 'ent)i3et*, 1o)'e m*2 %e'ome +)e*te) +)e*te) t(*n t(e 1o)'e o1 o1 1)i'tion. T(-s it is (*)e) (*)e) 1o) * '*) to t-)n )o-n * 'o)ne) 'o)ne) *t * (i+( s3ee. s3ee. . =16! T(e e"#erimet$l $l%e o& ' o%l+ e 're$ter $t #ole) t($ $t t(e e%$tor* ee i& t(e e$rt( ill e $ #er&ect )#(ere. E"#l$i> T(e 1o)'es *'tin+ on * %o2 sit-*te *t t(e e-*to) *)e t(e 1o)'e o1 +)*/it2 *n 'ent)i1-+*, 1o)'e. Hen'e t(e )es-,t*nt 1o)'e o)
F
m /= )
F< m +@ m ) ω= Hen'e6 t(e /*,-e o1 *''e,e)*tion *''e,e)*tion -e to +)*/it2 *t t(e e-*to)6 m +V< m + @ m ) ω= +V < + @ ) ω= . ω5 is t(e *n+-,*) /e,o'it2 o1 t(e e*)t(. T(-s t(e /*,-e o1 +5 wo-, %e +)e*te) *t 3o,es t(*n *t t(e e-*to). . =17! ?(, i) t(e $cceler$tio o& $ o+, moi' i $ circle +irecte+ to$r+) t(e cetre> "e'*-se6 *''o)in+ to t(e se'on ,*w o1 motion6 t(e i)e'tion o1 1o)'e is s*me *s t(e i)e'tion o1 *''e,e)*tion. T(e 'ent)i3et*, *''e,e)*tion is 3)o/ie %2 'ent)i3et*, 1o)'e t(*t is tow*)s t(e 'ent)e on 'i)',e. T(e)e1o)e6 t(e *''e,e)*tion o1 * %o2 mo/in+ in * 'i)',e 'i)',e i)e'te tow*)s t(e t(e 'ent)e. F' < m *' . =18! o o%l+ t(e ei'(t o& $ m$ e $&&ecte+ i& t(e e$rt( )to#) rot$ti' $o%t it) $"i) )%++el, W T(e )es-,t*nt 1o)'e *'tin+ on t(e m*n sit-*te *t t(e e-*to) tow*)s t(e 'ent)e o1 t(e e*)t( is +i/en6 m /= # < m + @ )
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
I1 t(e e*)t( sto3s )ot*tin+ s-en,2 t(e 'ent)i1-+*, 1o)'e *'tin+ on (im wi,, %e'ome 7e)o *n its wei+(t wi,, %e m?B m + < m + @ m + < m + ) .T(-s wei+(t o1 t(e m*n wi,, in')e*se. . =19! ?(, $re t(e e$r )i'(t) o& $ lo'Fr$'e ri&le $+j%)t$le> T(e)e *)e two si+(ts on t(e %*))e, o1 )i1,e. Ne*) si+(t is ',ose o e2e o1 s(oote) *n t(e 1*) si+(t is *t t(e en o1 t(e %*))e,. in'e (e )*n+e o1 %-,,et e3ens on t(e *n+,e *t w(i'( it is 1i)e. T(e ne*) si+(t '*n %e *-ste to '(*n+e t(e *n+,e o1 1i)in+. T(e 1i)in+ 1o)m t(e )i1,e is t(e e;*m3,e o1 3)oe'ti,e motion. T(-s ,on+@)*n+e )i1,e t(e ne*) si+(ts '*n %e *-ste *''o)in+ to t(e ist*n'e o1 t(e t*)+et. . =20! o +oe) $ir &rictio $&&ect) t(e #$t( o& $ #rojectile. Ai) 1)i'tion *11e'ts on t(e (o)i7ont*, /e,o'it2 'om3onent in 3)oe'ti,e motion. T(is 'om3onent m*2 in')e*ses o) e')e*ses6 e3ens on t(e i)e'tion o1 win %,owin+ . = 21! T(e ertic$l elocit, o& #rojectile #rojectile 'oe) o icre$)i' i t(e +o$r+ +irectio %t it) (oriBot$l elocit, rem$i) co)t$t. ?(,> "e'*-se6 t(e /e)ti'*, /e,o'it2 'om3onent6 o1 3)oe'ti,e is -ne) t(e *'tion o1 +)*/it2 w(i'(6 +oes on in')e*sin+ in own w*) motion i)e'tion %-t its (o)i7ont*, /e,o'it2 )em*ins 'onst*nt %2 ne+,e'tin+ *i) 1)i'tions. . =22! Doe) t(e (oriBot$l elocit, o& #rojectile rem$i e"$ctl, co)t$t i $ir* $l)o> I1 *i) *n ot(e) 1o)'e o1 1)i'tions *)e *)e not ne+,e'te ne+,e'te t(en t(e (o)i7ont*, /e,o'it2 /e,o'it2 'om3onent o1 3)oe'ti,e motion e;*'t,2 oes not )em*in 'onst*nt. !o; ≠ !; . =23! S(o t($t $ #rojectile c$ ($e )$me r$'e &or to $'le) o& #rojectio). T(e )*n+e 1o) two *n+,e o1 3)oe'tions *)e s*me t(ese *)e ,*-n'( *n+,e ?s*26 4 o wit( /e)ti'*, /e)ti'*, *n t*8e o11 *n+,e wit( = / R m*; < o + (o)i7ont*,. . =24! De&ie oe r$+i$. T(e 'i)'-,*) ,en+t( o1 * 'i)',e e-*, to t(e )*i-s o1 'i)',e is '*,,e one )*i*n5. . =25 Gie t(e t(e rel$tio etee etee $< $'%l$r $'%l$r $+ lie$r lie$r elocitie) elocitie) < $'%l$r $cceler$tio $+ $+ lie$r $'%l$r elocit, $+ time #erio+. :. / < ) ω 6 +i/es t(e )e,*tion %etween ,ine*) *n *n+-,*) /e,o'ities. =. * < ) α6 +i/es t(e )e,*tion %etween ,ine*) *n *n+-,*) *''e,e)*tion.
T< .
= Q 6 +i/es t(e )e,*tion %etween time 3e)io *n *n+-,*) /e,o'it2.
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
$cceler$tio c<
An o%e't mo/in+ in * 'i)',e is *''e,e)*tin+0 s-'( o%e'ts *)e '(*n+in+ t(ei) /e,o'it2 @ eit(e) t(e s3ee ?i.e.6 m*+nit-e o1 t(e /e,o'it2 /e'to) o) t(e i)e'tion. An o%e't -ne)+oin+ -ni1o)m 'i)'-,*) motion is mo/in+ wit( * 'onst*nt s3ee. An o%e't is *''e,e)*tin+ -e to its '(*n+e in i)e'tion. T(e i)e'tion o1 t(e *''e,e)*tion is inw*)s. T(e net 1o)'e *'tin+ -3on s-'( *n o%e't is i)e'te tow*)s t(e 'ente) o1 t(e 'i)',e. T(e net 1o)'e is s*i to %e *n inw*) o) 'ent)i3et*, 1o)'e. T(e net 1o)'e i)e'te 3e)3eni'-,*) to t(e /e,o'it20 t(e o%e't is *,w*2s '(*n+in+ its i)e'tion *n -ne)+oin+ * 'ent)i3et*, *''e,e)*tion. :. =. . 4. . K. . . . :B. ::. :=. :. :4. :.
:K.
:.
A'le9 t(e s3*'e %etween two ,ines o) 3,*nes t(*t inte)se't0 t(e in',in*tion o1 one ,ine to *not(e)0 me*s-)e in e+)ees Ac%te $'le* $'le* one ,ess t(*n * )i+(t *n+,e6 o) ,ess t(*n B e+)ees. A+j$cet or A+j$cet or Coti'%o%) $'le)* $'le)* s-'( *s (*/e one ,e+ 'ommon to %ot( *n+,es. Ri'(t $'le* $'le* one 1o)me %2 * )i+(t ,ine 1*,,in+ on *not(e) 3e)3eni'-,*),26 o) *n *n+,e o1 B e+)ees. Soli+ $'le* $'le* t(e 1i+-)e 1o)me %2 t(e meetin+ o1 t()ee o) mo)e 3,*ne *n+,es *t one 3oint. S#(eric$l $'le* $'le* one m*e %2 t(e meetin+ o1 two *)'s o1 +)e*t 'i)',es6 w(i'( m-t-*,,2 '-t one *not(e) on t(e s-)1*'e o1 * +,o%e o) s3(e)e. Iter$l $'le)* $'le)* t(ose w(i'( *)e wit(in *n2 )i+(t@,ine 1i+-)e. i"tilie$l $'le* $'le* one 1o)me %2 * )i+(t ,ine wit( * '-)/e ,ine. li%e $'le6 one *'-te o) o%t-se6 in o33osition to * )i+(t *n+,e. t%)e $'le* $'le* one +)e*te) t(*n * )i+(t *n+,e6 o) mo)e t(*n B e+)ees. A'%l$r +i)t$ce9 t(e *n+-,*) se3*)*tion %etween two o%e'ts *s 3e)'ei/e %2 *n o%se)/e)0 Z(e )e'o)e *n+-,*) ist*n'es %etween t(e st*)sZ. A'%l$r #oit* #oit* t(e 3oint *t w(i'( t(e sies o1 t(e *n+,e meet0 t(e /e)te;. A'%l$r motio* motio* t(e motion o1 * %o2 *%o-t * 1i;e 3oint o) 1i;e *;is6 *s o1 * 3,*net o) 3en-,-m. It is e-*, to t(e *n+,e 3*sse o/e) *t t(e 3oint o) *;is % 2 * ,ine )*wn to t(e %o2. %o 2. A'%l$r elocit,* elocit,* t(e )*tio o1 *n+-,*) motion to t(e time em3,o2e in es')i%in+. *n+,e *t w(i'( t(e ,e*st (ei+(t is ,ost in 3)o3o)tion to t(e +)o-n 'o/e)e. ?= T(e #tim%m A'le! ?: T(e +,iin+ *n+,e *t *n+,e o1 *tt*'8 o1 o1 *n *e)o1oi, *e)o1oi, *t *t w(i'( t(e )*tio o1 ,i1t ,i1t to to )*+ )*+ is is +)e*test. T(e *n+,e is s,i+(t,2 ,ess t(*n 4 e+)ees. E;*m3,e9 -33ose t(e o%e't is t()own = 1eet *%o/e t(e +)o-n wit( *n initi*, s3ee o1 .B 1tse'on. A 4 e+)ee *n+,e )es-,ts in t(e o%e't +oin+ .=4 1eet. T(e ist*n'e is .4K 1eet *t t(e o3tim*, *n+,e o1 B e+)ees. A'%l$r elocit, is e1ine *s t(e time )*te o1 '(*n+e o1 *n+-,*) is3,*'ement wit( )es3e't to * 3oint9 An+-,*) 1)e-en'2 Q ?*,so )e1e))e to %2 t(e te)ms *n+-,*) s3ee6 )*i*, 1)e-en'26 'i)'-,*) 1)e-en'26 1)e-en'26 *n )*i*n 1)e-en'2 is * s'*,*) me*s-)e o1 )ot*tion )*te. An+-,*) 1)e-en'2 1)e-en'2 ?o) *n+-,*) s3ee is t(e m*+nit-e o1 t(e /e'to) -*ntit2 A'%l$r elocit,. elocit,. T(e te)m *n+-,*) 1)e-en'2 /e'to) is sometimes -se *s * s2non2m 1o) t(e /e'to) -*ntit2 *n+-,*) /e,o'it2 . /e,o'it2 .
C) o Circ%l$r otio :.
#(i'( #(i'( one o1 t(e t(e 1o,,owin 1o,,owin+ + st*tements st*tements *%o-t 'i)''i)'-,*) ,*) motion motion is 'o))e 'o))e'tW 'tW A. T(e s3ee s3ee is '(*n+ '(*n+in+ in+ *,, *,, t(e t(e time. time. ". T(e ,ine*) ,ine*) s3ee s3ee is 'onst*nt 'onst*nt %-t %-t t(e *''e,e) *''e,e)*tion *tion is '(*n+in '(*n+in+ + *,, t(e time. time. C. T(e ,ine*) ,ine*) s3ee s3ee is 'onst* 'onst*nt nt *n t(e t(e /e,o'it2 /e,o'it2 is 'onst* 'onst*nt. nt. D. T(e ,ine*) s3ee s3ee is 'onst*nt 'onst*nt %-t t(e)e is *''e,e)*tion *''e,e)*tion %e'*-se t(e /e,o'it2 is '(*n+in+ *,, t(e time6 sin'e t(e)e t(e)e is * 'onst*nt '(*n+e in i)e'tion.
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#(i'( #(i'( one is 'o))e 'o))e't 't *%o-t *%o-t t(e 'ent)i3et 'ent)i3et*, *, 1o)'e 1o)'eWW A. It is in/e)se in/e)se,2 ,2 3)o3o)ti 3)o3o)tion*, on*, to to t(e /e,o'i /e,o'it2. t2. ". It *,w*2s *'ts *t B e+)ees e+)ees to t(e /e,o'it26 *n *w*2 1)om t(e 'ent)e o1 t(e 'i)',e 'i)',e C. It *,w*2s *,w*2s *'ts *t B e+)ees e+)ees to t(e /e,o'it2 /e,o'it266 *n tow*)s tow*)s t(e 'ent)e 'ent)e o1 t(e 'i)',e 'i)',e D. It is %*,*n'e %*,*n'e %2 t(e t(e 'ent)i 'ent)i1-+*, 1-+*, 1o)'e 1o)'e
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A moe, moe, *e)o 3,*ne 3,*ne is tet(e)e tet(e)e to * 3ost *n (e, %2 * 1ine ,ine. It 1,ies in * (o)i7ont*, 'i)',e. 'i)',e. T(en T(en t(e ,ine %)e*8s. %)e*8s. #(*t i)e'tion wi,, it 1,2 inW A. Di)e't,2 Di)e't,2 to t(e t(e 'ent)e 'ent)e o1 t(e 'i)',e 'i)',e.. ". Di)e't,2 Di)e't,2 *w*2 *w*2 1)om t(e 'ent)e 'ent)e o1 t(e t(e 'i)',e 'i)',e.. C. In * 'i)'-, 'i)'-,*) *) 3*t( 3*t(66 *s %e1o)e %e1o)e D. In * st)* st)*i+( i+(tt ,ine ,ine *t * t*n+en t*n+ent. t.
4.
A s*te s*te,,i ,,ite te is in in o)%it. o)%it. T(is T(is is is %e'*%e'*-se9 se9 A. T(e) T(e)ee is no no +)*/ +)*/it2 it2.. ". T(e s*te,,i s*te,,ite te is %ein+ %ein+ (e, t(e)e t(e)e %2 'ent)i1-+ 'ent)i1-+*, *, 1o)'e. 1o)'e.
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
C. G)*/it2 G)*/it2 is is *'tin+ *'tin+ *s * 'ent)i3et* 'ent)i3et*,, 1o)'e. 1o)'e. D. It is %ein+ %ein+ *tt)*'t *tt)*'te e %2 t(e +)*/it2 +)*/it2 o1 ot(e) ot(e) o%e'ts o%e'ts in in s3*'e. s3*'e.
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Cent)i3et* Cent)i3et*,, 1o)'e 1o)'e '*n '*n %e wo)8e wo)8e o-t o-t wit( wit( t(e )e,*tions )e,*tions(i3 (i3 A. F < /[=) ". F < m/[== C. F < m/[=) D. F < m)[=/ E. #(i'( #(i'( one o1 t(e 1o,,owi 1o,,owin+ n+ st*tements st*tements is is 'o))e't 'o))e't w(en * m*ss m*ss is m*e to to mo/e in * /e)ti'*, /e)ti'*, 'i)',e 'i)',e A. T(e tension in t(e st)in+ st)in+ is +)e*test +)e*test *t t(e %ottom o1 t(e 'i)',e *n ,e*st w(en w(en *t t(e to3. ". T(e tension tension in t(e t(e st)in+ is +)e*te +)e*test st *t t(e to3 o1 o1 t(e 'i)',e 'i)',e *n ,e*st ,e*st w(en w(en *t t(e %ottom. %ottom. C. T(e tension tension is 3)o/ie 3)o/ie %2 * 'om%in*tio 'om%in*tion n o1 'ent)i1-+ 'ent)i1-+*, *, 1o)'e 1o)'e *n wei+(t. wei+(t. D. T(e wei+(t wei+(t *'ts in in * i)e'tion i)e'tion *w*2 *w*2 1)om t(e t(e 'ent)e 'ent)e o1 t(e 'i)',e. 'i)',e.
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An+-,* An+-,*)) /e, /e,o'i o'it2 t2 is me*sme*s-)e )e in9 A. Mete Mete)s )s 3e) 3e) se' se'on on ". De+) De+)ee eess 3e) 3e) se'o se'on n C. No -nit -nitss *t *t *,, *,, D. R*i R*i*n *nss 3e) 3e) se'o se'on n
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An+-,*) An+-,*) /e,o'it2 /e,o'it2 is -se -se in 3)e1e)en 3)e1e)en'e 'e to s3ee s3ee %e'*-se %e'*-se99 A. T(e s3ee s3ee e3en e3enss on t(e t(e )*i)*i-s. s. ". T(e 'ent)i 'ent)i3et*, 3et*, 1o)'e 1o)'e *'ts *'ts *t B e+)e e+)ees es to t(e s3ee. s3ee. C. T(e s3ee s3ee oes oes not '(*n+e '(*n+e.. D. 3ee 3ee is is (*)e (*)e)) to me*s-) me*s-)e. e.
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A '*) '*) +oes +oes *)o-n *)o-n * %en. %en. #(*t #(*t 3)o/ies 3)o/ies t(e t(e 'ent)i3e 'ent)i3et*, t*, 1o)'eW 1o)'eW A. G)*/it2 ". F)i'tion C. E,e't) E,e't)ost ost*ti *ti'' *tt)* *tt)*'ti 'tion on D. No 1o)'e o)'e *t *,,. *,,.
:B. A +i), o1 m*ss B 8+ is on * )o-n*%o-t o1 )*i-s m t(*t m*8es one 'om3,ete 'om3,ete t-)n e/e)2 se'ons. #(*t #(*t is t(e 'ent)i3et*, 1o)'e *'tin+ on (e)W A. N ". 4 N C. =K N D. BB N
Prof: Najeeb Mughal, Edited by Tarvesh Kumar
Fo) Mo)e Notes O1 C(emist)26 P(2si's6 P(2si's6 En+,is( *n Ot(e) -%e'ts !isit9
