The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. (a) Compute the acceleration due to gravity on the surface of Venus from these data.

Verified step by step guidance

1. Start by recalling the formula for gravitational acceleration on the surface of a planet: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>g</mi> = <mfrac><mrow><mi>G</mi><mi>M</mi></mrow><msup><mi>R</mi><mn>2</mn></msup></mfrac></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>G</mi></math> is the gravitational constant, <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>M</mi></math> is the mass of the planet, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>R</mi></math> is the radius of the planet.
2. Express the mass of Venus in terms of Earth's mass: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>M</mi><msub><mi>v</mi></msub> = 0.815 <mi>M</mi><msub><mi>e</mi></msub></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>M</mi><msub><mi>e</mi></msub></math> is the mass of Earth.
3. Express the radius of Venus in terms of Earth's radius: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>R</mi><msub><mi>v</mi></msub> = 0.949 <mi>R</mi><msub><mi>e</mi></msub></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>R</mi><msub><mi>e</mi></msub></math> is the radius of Earth.
4. Substitute the expressions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>M</mi><msub><mi>v</mi></msub></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>R</mi><msub><mi>v</mi></msub></math> into the gravitational acceleration formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>g</mi><msub><mi>v</mi></msub> = <mfrac><mrow><mi>G</mi><mo>(</mo>0.815 <mi>M</mi><msub><mi>e</mi></msub><mo>)</mo></mrow><msup><mrow><mo>(</mo>0.949 <mi>R</mi><msub><mi>e</mi></msub><mo>)</mo></mrow><mn>2</mn></msup></mfrac></math>.
5. Simplify the expression to find the acceleration due to gravity on Venus: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>g</mi><msub><mi>v</mi></msub> = <mfrac><mrow>0.815 <mi>G</mi><mi>M</mi><msub><mi>e</mi></msub></mrow><mrow>0.949<msup><mi>R</mi><msub><mi>e</mi></msub><mn>2</mn></msup></mrow></mfrac></math>. Use the known values of <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>G</mi></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>M</mi><msub><mi>e</mi></msub></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>R</mi><msub><mi>e</mi></msub></math> to compute the numerical value.

Key Concepts

Gravitational Acceleration

Gravitational acceleration is the acceleration of an object due to the force of gravity acting on it. It is calculated using the formula g = G * M / R^2, where G is the gravitational constant, M is the mass of the celestial body, and R is its radius. This formula helps determine how strong gravity is on the surface of a planet like Venus.

Proportionality and Ratios

Proportionality and ratios are mathematical concepts used to compare quantities. In this context, they help relate the mass and radius of Venus to those of Earth. By understanding that Venus's mass is 81.5% and its radius is 94.9% of Earth's, we can use these ratios to adjust Earth's gravitational acceleration to find that of Venus.

Earth's Gravitational Acceleration

Earth's gravitational acceleration is approximately 9.81 m/s². This value serves as a reference point for calculating the gravitational acceleration on other planets. By knowing the relative mass and radius of Venus compared to Earth, we can adjust this value to estimate the gravitational acceleration on Venus's surface.