In a gravitational field, if a body is bound with earth, then total mechanical energy is

Options

A. positive
B. zero
C. negative
D. may be positive, negative or zero

Explanation

To determine the total mechanical energy of a body bound to the Earth in a gravitational field, we can follow these steps:

### Step-by-Step Solution

1. **Understanding Total Mechanical Energy**:
The total mechanical energy (E) of a system is the sum of its kinetic energy (K) and potential energy (U). For a body in a gravitational field, this can be expressed as:
\[ E = K + U \]

2. **Potential Energy in a Gravitational Field**:
The gravitational potential energy (U) of a body of mass \( m \) at a distance \( r \) from the center of the Earth (mass \( M \)) is given by the formula:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the universal gravitational constant.

3. **Kinetic Energy**:
The kinetic energy (K) of the body can be expressed as:
\[ K = \frac{1}{2}mv^2 \]
where \( v \) is the velocity of the body.

4. **Bound System**:
For a body to be bound to the Earth, it must have a total mechanical energy that is negative. This means that the kinetic energy must be less than the magnitude of the potential energy, leading to:
\[ E < 0 \]

5. **Condition at Infinity**:
If we consider the body being moved to an infinite distance from the Earth, the gravitational potential energy approaches zero:
\[ U \to 0 \quad \text{as} \quad r \to \infty \]
At this point, the kinetic energy would also be zero if the body is at rest at infinity.

6. **Conclusion**:
Since the total mechanical energy must be negative for the body to remain bound to the Earth, we conclude that:
\[ E < 0 \]
Therefore, the total mechanical energy of a body bound to the Earth in a gravitational field is negative.

### Final Answer
The total mechanical energy of a body bound to the Earth in a gravitational field is negative.

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