The weight of an object is 90 kg at the surface of the earth. If it is taken to a height equal to half of the radius of the earth, then its weight will become

Options

A. 135 kg
B. 45 kg
C. 60 kg
D. 40 kg

Explanation

To solve the problem of finding the weight of an object at a height equal to half the radius of the Earth, we can follow these steps:

### Step 1: Understand the Weight Change Formula
The weight of an object at a height \( h \) above the surface of the Earth can be calculated using the formula:
\[ W' = \frac{W}{(1 + \frac{h}{R})^2} \]
where:
- \( W' \) is the weight at height \( h \),
- \( W \) is the weight at the surface of the Earth,
- \( R \) is the radius of the Earth,
- \( h \) is the height above the surface.

### Step 2: Identify Given Values
From the problem, we know:
- The weight at the surface of the Earth, \( W = 90 \, \text{kg} \) (Note: This is actually the mass, but we will treat it as weight in this context).
- The height \( h \) is equal to half the radius of the Earth, so \( h = \frac{R}{2} \).

### Step 3: Substitute Values into the Formula
Substituting \( h = \frac{R}{2} \) into the weight change formula:
\[ W' = \frac{W}{(1 + \frac{R/2}{R})^2} \]
This simplifies to:
\[ W' = \frac{W}{(1 + \frac{1}{2})^2} = \frac{W}{(1.5)^2} = \frac{W}{2.25} \]

### Step 4: Calculate the New Weight
Now, substituting \( W = 90 \, \text{kg} \):
\[ W' = \frac{90}{2.25} \]
Calculating this gives:
\[ W' = 40 \, \text{kg} \]

### Conclusion
Thus, the weight of the object at a height equal to half the radius of the Earth will be \( 40 \, \text{kg} \).

### Final Answer
The weight of the object at the specified height is **40 kg**.

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