Solution:

Given data:
- Diameter of the drum (D) = 80 cm
- Length of the drum (L) = 3 m
- Shell thickness (t) = 1 cm
- Internal pressure (P) = 2.5 N/mm^2
- Modulus of elasticity (E) = 2 x 10^5 N/mm^2
- Poisson's ratio (v) = 0.25

We need to find the change in diameter of the drum due to the internal pressure.

Calculations:

1. First, we need to find the hoop stress (σh) in the drum due to internal pressure.
- Hoop stress is the stress that acts circumferentially in the shell of the drum.
- It can be calculated using the formula:
σh = PD/2t
where,
P = Internal pressure
D = Diameter of the drum
t = Shell thickness

Substituting the given values, we get:
σh = (2.5 x 80)/(2 x 1) = 100 N/mm^2

2. Next, we need to find the longitudinal stress (σl) in the drum due to internal pressure.
- Longitudinal stress is the stress that acts longitudinally (along the length) in the shell of the drum.
- It can be calculated using the formula:
σl = PD/4t
where,
P = Internal pressure
D = Diameter of the drum
t = Shell thickness

Substituting the given values, we get:
σl = (2.5 x 80)/(4 x 1) = 50 N/mm^2

3. Now, we can use the equation for change in diameter (ΔD) of a thin cylindrical shell due to internal pressure, which is given by:
ΔD/D = (σh - vσl)/E
where,
ΔD = Change in diameter
D = Original diameter
σh = Hoop stress
σl = Longitudinal stress
v = Poisson's ratio
E = Modulus of elasticity

Substituting the values, we get:
ΔD/80 = (100 - 0.25 x 50)/(2 x 10^5)
ΔD = 0.000375 cm

Therefore, the change in diameter of the drum due to internal pressure is 0.000375 cm or 0.00375 mm.

Conclusion:

- The change in diameter of the drum due to internal pressure is found to be 0.000375 cm or 0.00375 mm.
- The calculations are based on the given data and the equations for hoop stress, longitudinal stress, and change in diameter of a thin cylindrical shell due to internal pressure.
- These calculations are important in designing and analyzing pressure vessels, pipes, and other cylindrical structures that are subjected to internal or external pressure.

0.035cm