2018-10-01_Physics_For_You

6. (b) : The CM of the rod has fallen

by

l

2

(1 – cos q).

? 'U = – Mg l

2

(c 1  os )









 ;

'K =^1

2

IZ^2 =

1

23

2

Ml 2









.

'U + 'K = 0 or 'K = –'U

Ÿ

1

23

Ml^22









. = Mg

l

2

 .(c 1  os )









Z^2 =

31 g

l

(cos )

Z^2 =

6

2

g^2

l

sin







 Ÿ Z =^6

2

g

l

.sin











7. (c) : Torque due to F must
   be more than torque due
   to mg about P.
   Fmin(r – h) = mgx
   Fmin(r – h)

= mg rr^22 ()h

Fmin =

mg rrhrh

rh

(^22)  (^2)  2


()

()

=

mg hrh

rh

()

()

2 



.

8. (c) : For string to be wound around the nail,

vgt 5 r

Ÿ vg 51 () 0 L

Now

1

2

mv^2 = mg(10) Ÿ v^2 = 2g (10)

Ÿ (^21) gg. 05 () 10 L
20 g > 5g(10 – L) Ÿ 20 > 50 – 5L Ÿ 5 L > 30
L > 6
Lmin = 6 cm.

9. (a) : Mass of square plate M 1 = l^2 V

Mass of circular disc M 2 = πσ

l^2

4

l/2 l/

l/2 l/2 cos q

q

F

r

P

x h

mg

x

M ll

l

CM=

×+

+









1

2

2

0

4

1

4

πσ

σ

π

Ÿ xCM= ll

+

=

+

π

π

π

4 4 π

4

() 4

xCM<l

2

, i.e., COM lies in the square

10. (a) : Acceleration of system

a

mg mg

m

=

sins 60 °− in 30 °

2

Here m = mass of each block

or ag=

 −







31

4

Now 



a

ma ma

CM m

=

12 +

2

Here

a

1 and



a 2 are^31

4

 −





g at the right angles.

Hence, aaCM== g

 −







2

2

31

42

11. (a) : If v is the velocity by which balls strike the
    bases, then their velocities aer collision are 0.6 v
    and 0.4 v as shown in the gure ‘u’ is the velocity
    of COM just aer collision and Z is the angular
    velocity acquired by the rod.

A B

0.6 v u 0.4 v

Z

For end A : 0.6v = u

+l

2

.ω ...(i)

For end B : 0.4v = u

l

−

2

.ω ...(ii)

Subtracting (i) and (ii) we get

0.2v = Zl Ÿ ω=

v

5 l

Now using [v^2 = u^2 + 2as]

v^2 = 0 + 2 × 10 × 0.2 = 4 Ÿ v = 2 m s–

ω=

×

= −

2

51

2

5

rad s^1

12. (d) : We can observe each
    and every element of rod in
    rotating with dierent radius
    about the axis of rotation.
    Take an elementary mass dm
    of the rod.

l

r

dl

q

dm