Problem Analysis

To calculate the minimum push required to roll the oil drum, we need to consider the following factors:

1. Weight of the drum: Since the weight of the drum is neglected, it does not affect the minimum push required.
2. Height of the footstep: The height of the footstep will determine the change in potential energy of the drum as it rolls over it.
3. Diameter and length of the drum: The dimensions of the drum will determine the moment of inertia of the drum and its resistance to rolling.

Solution

1. Calculate the change in potential energy:
- The change in potential energy of the drum can be calculated using the formula: ΔPE = m * g * h, where m is the mass of the drum, g is the acceleration due to gravity, and h is the height of the footstep.
- Since the density of oil is given as 1 kg/L, we can calculate the mass of the drum using its volume: V = π * r^2 * h, where r is the radius of the drum and h is its length.
- Substituting the given values, we have V = π * (0.5 m)^2 * 1.5 m = 1.1781 m^3.
- Since the density of oil is 1 kg/L, the mass of the drum is calculated as m = V * ρ = 1.1781 m^3 * 1000 kg/m^3 = 1178.1 kg.
- Substituting the values into the change in potential energy formula, we have ΔPE = 1178.1 kg * 9.81 m/s^2 * 0.1 m = 1152.38 J.

2. Calculate the moment of inertia:
- The moment of inertia of the drum can be calculated using the formula: I = 0.5 * m * r^2, where m is the mass of the drum and r is the radius of the drum.
- Substituting the given values, we have I = 0.5 * 1178.1 kg * (0.5 m)^2 = 0.1473 kg*m^2.

3. Calculate the minimum push required:
- The minimum push required to roll the drum can be calculated using the formula: P = ΔPE / (r + 0.5 * h), where ΔPE is the change in potential energy, r is the radius of the drum, and h is the height of the footstep.
- Substituting the given values, we have P = 1152.38 J / (0.5 m + 0.5 * 0.1 m) = 1444.47 N.

Conclusion

The minimum push required at the top of the drum to roll it across a footstep of 100 mm high is 1444.47 N.