Anvils made of single crystals of diamond, with the shape as shown in Figure, are used to investigate the behaviour of materials under very high pressures. Flat faces at the narrow end of the anvil have a diameter of 0.50 mm, and the wide ends are subjected to a compressional force of 50,000 N. What is the pressure at the tip of the anvil?

#### Solution 1

Diameter of the cones at the narrow ends, *d* = 0.50 mm = 0.5 × 10^{–3} m

Radius, `r = d/2 = 0.25 xx 10^(-3) m`

Compressional force,* F* = 50000 N

Pressure at the tip of the anvil:

`p = "Force"/("Area"pi) = F/r^2`

`= 50000/(pi(0.25 xx 10^(-3))^2)`

`= 2.55 xx 10^11 Pa`

Therefore, the pressure at the tip of the anvil is 2.55 × 10^{11} Pa.

#### Solution 2

Diameter of the corner end of the anvil

`d = 0.50 mm = 0.50 xx 10^(-3) m`

Area of cross-section of tip,

`A = (pid^2)/4`

`= (22xx(0.50 xx 10^(-3))^2)/(7xx4) m^2`

Stress (= pressure at the tip of the anvil)

= `F/A = (50000 xx 4 xx 7)/(22 xx (0.50)^2 xx 10^(-6)) Nm^(-2)`

` 2.54 xx 10^11 Nm^(-2) (or Pa)`