Final answer:
The acceleration of gravity at the surface of a world with five times Earth's mass and seven times its radius would be approximately 1.002 m/s².
Step-by-step explanation:
To find what would be the acceleration of gravity (in m/s²) at the surface of a world with five times Earth's mass and seven times its radius, we use Newton's Universal Law of Gravitation. The acceleration due to gravity, g, at the surface of a planet is given by the formula:
g = G * M / R²
where G is the gravitational constant (6.674×10⁻¹¹ m³kg⁻¹s⁻²), M is the mass of the planet, and R is the planet's radius. On Earth, the acceleration due to gravity is approximately 9.8 m/s². Taking Earth's mass and radius as the base values, the mass of this hypothetical planet is 5 times Earth's mass (5M) and its radius is 7 times Earth's radius (7R). Plugging these values into the formula gives:
g = G * 5M / (7R)² = G * 5M / 49R²
Since Earth's gravity (g) is G * M / R², we can deduce that the new acceleration due to gravity is:
g = (1/49) * 5 * Earth's gravity = (5/49) * 9.8 m/s²
g ≈ 1.002 m/s² (rounded to three decimal places).