In a system on Earth, the total mechanical energy of an object can be zero if the kinetic energy and potential energy are equal in magnitude but opposite in sign.
The total mechanical energy (E) of an object is the sum of its kinetic energy (KE) and potential energy (PE). Mathematically, it can be represented as:
E = KE + PE
Kinetic energy (KE) is given by:
KE = (1/2)mv^2
where m is the mass of the object and v is its velocity.
Potential energy (PE) depends on the type of potential energy involved. For example, for an object at a height h above the ground, the potential energy due to gravity is given by:
PE = mgh
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, let's consider a situation where an object is moving upward against the force of gravity. As it moves upward, its potential energy increases, while its kinetic energy decreases because its velocity decreases. At a certain point, if the object reaches a height where its potential energy exactly balances out the decrease in kinetic energy, the total mechanical energy can be zero.
In this case, the potential energy is positive (indicating the object's position above a reference point), and the kinetic energy is negative (since the velocity is decreasing). The magnitudes of the potential energy and kinetic energy are equal but opposite in sign, resulting in a total mechanical energy of zero.
It's important to note that this is a specific scenario and doesn't represent the typical behavior of objects in most situations. In general, objects on Earth will have non-zero total mechanical energy due to the effects of gravity and motion.