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

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

### Step 2: Defining Potential Energy
The gravitational potential energy (P.E.) of a body of mass \( m \) at a distance \( r \) from the center of the Earth (with mass \( M \)) is given by the formula:
\[ P.E. = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant.

### Step 3: Defining Kinetic Energy
The kinetic energy (K.E.) of a body moving with velocity \( v \) is given by:
\[ K.E. = \frac{1}{2} mv^2 \]

### Step 4: Total Mechanical Energy in a Bound System
For a body that is bound to the Earth, the total mechanical energy is negative. This indicates that the body is in a stable orbit or bound state. The total mechanical energy can be expressed as:
\[ E = K.E. + P.E. \]

### Step 5: Using the Relationship Between K.E. and P.E.
In a bound gravitational system, the kinetic energy and potential energy are related. For a circular orbit, it can be shown that:
\[ K.E. = -\frac{1}{2} P.E. \]
Thus, substituting the potential energy into the equation gives:
\[ E = K.E. + P.E. = -\frac{1}{2} P.E. + P.E. = -\frac{1}{2} \left(-\frac{G M m}{r}\right) + \left(-\frac{G M m}{r}\right) \]
This simplifies to:
\[ E = -\frac{G M m}{2r} \]

### Step 6: Conclusion
Since the total mechanical energy \( E \) is negative, we conclude that the body is indeed bound to the Earth.

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

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