A homologous rod of mass M=20kg of length L=3m and of negligible cross-section rests initially on the horizontal ground chosen as a reference level for the gravitational potential energy. The rod is rotated about one end until it takes a vertical position . Calculate the variation of the mechanical energy of the system (rod, earth).

the center of gravity moves up 1.5 meters

m g h
= 20 * 9.81 * 1.5

To calculate the variation of the mechanical energy of the system (rod, earth) as the rod is rotated from a horizontal to a vertical position, we need to consider the different forms of energy involved.

Let's break down the steps to find the variation of mechanical energy:

Step 1: Determine the potential energy of the rod-Earth system in the initial horizontal position.
The potential energy of an object near the Earth's surface is given by the formula: PE = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height of the object.

In this case, the rod is initially on the ground, so its height is zero. Therefore, the potential energy is also zero.

Step 2: Determine the potential energy of the rod-Earth system in the final vertical position.
When the rod is in the vertical position, its height is equal to its length.

So, the potential energy at the final position is: PE = m * g * L, where L is the length of the rod.

Step 3: Calculate the variation of the potential energy.
The variation of the potential energy is the difference between the potential energy in the final and initial positions.

Therefore, the variation of potential energy is: ΔPE = m * g * L - 0 = m * g * L.

Step 4: Determine the change in kinetic energy.
Since the rod is rotating, it will have rotational kinetic energy.

The formula for rotational kinetic energy is: KE_rotational = (1/2) * I * ω^2, where I is the moment of inertia and ω is the angular velocity.

For a thin rod rotating about one end, the moment of inertia is given by: I = (1/3) * m * L^2.

When the rod is in the vertical position, it rotates with an angular velocity of ω = 0, as it comes to rest.

Therefore, the initial kinetic energy is zero.

Step 5: Calculate the variation of mechanical energy.
The variation of mechanical energy is a sum of the variations of potential and kinetic energy.

ΔE_mechanical = ΔPE + ΔKE

Substituting the values we have, the variation of mechanical energy in this case is:

ΔE_mechanical = m * g * L + 0 = m * g * L.

Now, plugging in the given values, ΔE_mechanical = 20 kg * 9.8 m/s^2 * 3 m = 588 J.

Therefore, the variation of mechanical energy of the system (rod, Earth) is 588 Joules.