a particle (C) of mass m = 0.2 kg can slide without friction on an inclined plane making an angle alfa (a) = 30 degrees with respect to the horizontal (aka ground) . (C) is launched at at an instant to=0 along a line Ox of greatest slope with an initial velocity of Vo =5 m/s . thus , (C) moves along Ox. At any instant t , the position of (C) is given by its abscissa x=OA. the horizontal plane passing through O is the reference level for the gravitational potential.

a) calculate the mechanical energy of the system {(C) earth.
b) express in terms of x the gravitational potential energy of the system {(C) earth}

a) To calculate the mechanical energy of the system {(C) Earth}, we need to consider the kinetic energy and potential energy.

The kinetic energy of the particle (C) is given by the formula:

KE = (1/2) * m * V^2

where m is the mass of the particle (C) and V is its velocity.

In this case, m = 0.2 kg and V = 5 m/s. Plugging in these values, we can calculate the kinetic energy.

KE = (1/2) * 0.2 kg * (5 m/s)^2
= 0.5 kg * 25 m^2/s^2
= 12.5 Joules

Now let's calculate the potential energy. The potential energy is given by the formula:

PE = m * g * h

where m is the mass of the particle (C), g is the acceleration due to gravity, and h is the height difference between the reference level and the current position.

Since the particle is sliding on the inclined plane, the height difference h can be expressed as h = x * sin(α), where x is the abscissa position and α is the angle of the inclined plane.

In this case, α = 30 degrees and x is the abscissa position along the inclined plane.

Now, let's calculate the potential energy at any instant t. We plug in the values:

PE = 0.2 kg * 9.8 m/s^2 * x * sin(30 degrees)
= 1.96*x Joules

b) To express the gravitational potential energy of the system {(C) Earth} in terms of x, we multiply the mass of the particle (C) and the gravitational acceleration by the height difference h, which is given by x * sin(α).

So, the gravitational potential energy (PE) in terms of x is:

PE = m * g * h
= 0.2 kg * 9.8 m/s^2 * x * sin(30 degrees)
= 1.96*x Joules

Therefore, the gravitational potential energy in terms of x is 1.96x Joules.