Given:
Density of iron (ρ_iron) = 7.8 * 10^3 kg/m^3
Volume of iron piece (V) = 100 cm^3 = 100 * 10^-6 m^3
Density of water (ρ_water) = 1000 kg/m^3
Acceleration due to gravity (g) = 10 m/s^2

To calculate:
(i) Weight of iron piece in air
(ii) Upthrust
(iii) Apparent weight of iron piece in water

Solution:
(i) Weight of iron piece in air:
The weight of an object is given by the formula:
Weight (W) = Mass (m) * Acceleration due to gravity (g)

The mass of the iron piece can be calculated using the formula:
Mass (m) = Density (ρ) * Volume (V)

Substituting the given values:
Mass (m) = 7.8 * 10^3 kg/m^3 * 100 * 10^-6 m^3
= 7.8 kg

Weight (W) = 7.8 kg * 10 m/s^2
= 78 N

Therefore, the weight of the iron piece in air is 78 N.

(ii) Upthrust:
Upthrust is the buoyant force experienced by an object when it is immersed in a fluid. It is equal to the weight of the fluid displaced by the object.

The volume of the iron piece is given as 100 cm^3. To calculate the volume of water displaced by the iron piece, we need to convert the volume to m^3.

Volume (V_water) = 100 * 10^-6 m^3

The upthrust (U) can be calculated using the formula:
Upthrust (U) = Density of fluid (ρ_water) * Volume of fluid displaced (V_water) * Acceleration due to gravity (g)

Substituting the given values:
Upthrust (U) = 1000 kg/m^3 * 100 * 10^-6 m^3 * 10 m/s^2
= 1 N

Therefore, the upthrust experienced by the iron piece is 1 N.

(iii) Apparent weight of iron piece in water:
The apparent weight of an object in a fluid is the difference between its weight in air and the upthrust it experiences.

Apparent weight (W_apparent) = Weight in air (W) - Upthrust (U)

Substituting the given values:
Apparent weight (W_apparent) = 78 N - 1 N
= 77 N

Therefore, the apparent weight of the iron piece in water is 77 N.