A solid sphere has a radius 21 cm. It is melted to form a cube. 20% material is wasted in this process. The cube is melted to form hemisphere. In thisprocess 20% material is wasted. The hemisphere is melted to form two spheres of equal radius. 20% material was also wasted in this process. What is the radius (in cm) of each new sphere?

The ratio of total surface area and volume of a sphere is 1 : 7. This sphere is melted to form small spheres of equal size. The radius of each small sphere is1/6ththe radius of the large sphere. What is the sum (in cm2) of curved surface areas of all small spheres?

A

31276

B

36194

C

25182

D

33264

Three solid spheres have their radii r1, r2 and r3. The spheres are melted to form a solid sphere of bigger radius. Then the radius of the new sphere is

A

r1+r2+r3

B

r12+r22+r3212

C

r13+r23+r3313

D

r14+r24+r3414

10 identical solid spherical balls of radius 3 cm are melted to form a single sphere. In this process 20% of solid is wasted. What is the radius (in cm) ofthe bigger sphere?

A

24

B

12

C

8

D

6

Volume of a cube of side ‘$$l$$’ and volume of a sphere of radius ‘r’ are equal. Both the cube and the sphere are solid and of same material. They are being cast. The ratio of the solidification time of the cube to the same of the sphere is

A

$${\left( {\frac{{4\pi }}{6}} \right)^3}{\left( {\frac{{\text{r}}}{l}} \right)^6}$$

B

$$\left( {\frac{{4\pi }}{6}} \right){\left( {\frac{{\text{r}}}{l}} \right)^2}$$

C

$${\left( {\frac{{4\pi }}{6}} \right)^2}{\left( {\frac{{\text{r}}}{l}} \right)^3}$$

D

$${\left( {\frac{{4\pi }}{6}} \right)^2}{\left( {\frac{{\text{r}}}{l}} \right)^4}$$

Thousand solid metallic spheres of diameter 6 cm are melted and recast into a new solid sphere. The diameter of the new sphere (in cm) is

A

30

B

90

C

45

D

60

A hollow rod has frictionless inner walls. Inside the rod are two small spheres that are kept on either side of the centre of rod. The rod is rotated about its central axis which is perpendicular to the plane of rotation. If the rod had been rotated by an impulsive torque which gave it an instantaneous angular velocity of 5rad/s, what will be the angular velocity after some finite but long time ‘t’? Assume no external forces act on rod after the impulse. Also state what will happen to the spheres in the centre of the rod. The rod has a mass =2kg & length =10cm. The two spheres have a mass =1kg each.

A

2.5 rad/s, spheres will stay where they are as there is no friction to move them

B

0 rad/s, spheres will move outwards and decrease velocity of rod to zero

C

1.25 rad/s, spheres will move to opposite ends of rod

D

2.5 rad/s, spheres will move to opposite ends of rod

Two sphere of radius 2 cm each are melted and casted into a single sphere. What would be the radius of this new sphere?

A

4 cm

B

161/3 cm

C

121/3 cm

D

2 cm

A solid cone of height 24 cm and radius of its base 8 cm is melted to form a solid cylinder of radius 6 cm and height 6 cm. In the whole process what
percent of material is wasted?

A

48.5

B

37.5

C

57.8

D

64

A solid cone of height 36 cm and radius of base 9 cm is melted to form a solid cylinder of radius 9 cm and height 9 cm. What percent of material is wasted this process?

A

25

B

5

C

0

D

10

Consider a given volume of liquid at a temperature T below Tm with a free energy G’. If some of the atoms of the liquid cluster together to form a small sphere of solid, the free energy of the system will change to G, given by________ (Assume V1 volume of the solid sphere, V2 volume of liquid, A is the solid/liquid interfacial area, γ solid/liquid interface free energy, G1 and G2 are the free energies per unit volume of solid and liquid respectively).

A

V1G1 + V2G2 + Aγ

B

V1G2 + V2G1 + Aγ

C

V1G1 + V2G2 – Aγ

D

V1G2 + V2G1 – Aγ