Question

5 formula 1 point a small asteroid has a mass of 632,794,535 kg. the asteroid’s gravitational field strength is equal to 0.00000071 newtons per kilogram. calculate the radius of the asteroid in meters. you can assume the asteroid is a sphere. answer

Explanation:

Step1: Recall Gravitational Field Formula

The formula for gravitational field strength \( g \) at the surface of a sphere (asteroid, assumed spherical) is \( g = \frac{GM}{r^2} \), where \( G = 6.67430 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \) (gravitational constant), \( M \) is the mass of the asteroid, and \( r \) is the radius we need to find. Rearranging for \( r \), we get \( r = \sqrt{\frac{GM}{g}} \).

Step2: Identify Given Values

Given: \( M = 632794535 \, \text{kg} \), \( g = 0.00000071 \, \text{N/kg} = 7.1 \times 10^{-7} \, \text{N/kg} \), \( G = 6.67430 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).

Step3: Substitute Values into Formula

First, calculate \( GM \):
\( GM = (6.67430 \times 10^{-11}) \times (632794535) \).
\( 632794535 \approx 6.32794535 \times 10^8 \), so \( GM \approx 6.67430 \times 10^{-11} \times 6.32794535 \times 10^8 \).
\( GM \approx 6.67430 \times 6.32794535 \times 10^{-3} \approx 42.24 \times 10^{-3} = 0.04224 \, \text{N m}^2/\text{kg} \).

Then, divide by \( g \):
\( \frac{GM}{g} = \frac{0.04224}{7.1 \times 10^{-7}} \approx \frac{0.04224}{7.1} \times 10^7 \approx 0.005949 \times 10^7 = 59490 \).

Step4: Take Square Root

\( r = \sqrt{59490} \approx 243.9 \approx 244 \, \text{meters} \) (rounded to a reasonable precision). Wait, let's recalculate more accurately:

First, \( GM = 6.67430 \times 10^{-11} \times 632794535 \):
\( 632794535 \times 6.67430 \times 10^{-11} = 632794535 \times 6.67430 \div 10^{11} \).
\( 632794535 \times 6.67430 \approx 632794535 \times 6 + 632794535 \times 0.67430 \approx 3796767210 + 426693300 \approx 4223460510 \).
Divide by \( 10^{11} \): \( 4223460510 \div 10^{11} = 0.0422346051 \, \text{N m}^2/\text{kg} \).

Now, \( \frac{GM}{g} = 0.0422346051 \div 7.1 \times 10^7 \) (since \( g = 7.1 \times 10^{-7} \), dividing by it is multiplying by \( 10^7/7.1 \)):
\( 0.0422346051 \times 10^7 = 422346.051 \); \( 422346.051 \div 7.1 \approx 59485.36 \).

Then \( r = \sqrt{59485.36} \approx 243.9 \approx 244 \, \text{meters} \) (or more precisely, using a calculator: \( \sqrt{59485.36} \approx 243.9 \), so approximately 244 meters).

Answer:

\boxed{244} (or more precise value like 243.9, depending on calculation precision)