You are an astronaut (m = 90 kg) and travel to a planet that is the same mass and size as Earth, but it has a rotational period of only 15 h. What is your apparent weight at the equator of this planet?

To determine your apparent weight at the equator of this planet, we need to consider the effect of rotation on gravity.

First, let's understand the concept of apparent weight. It refers to the force experienced by a person due to the combination of gravity and any upward or downward acceleration. In this case, we can assume that the astronaut is not moving vertically, so their apparent weight is essentially the force of gravity acting on them.

Now, to calculate the apparent weight at the equator of a planet with a rotational period of 15 hours, we need to use the formula for gravitational force:

F = (G * m1 * m2) / r^2,

where:
- F is the gravitational force,
- G is the gravitational constant (approximately 6.67430 x 10^-11 N m^2/kg^2),
- m1 and m2 are the masses of two objects (in this case, the astronaut and the planet),
- r is the distance between the centers of the two objects.

Since the planet is the same mass and size as Earth, we can assume that its mass (m2) is the same as Earth's mass, which is approximately 5.972 x 10^24 kg.

The rotational period of 15 hours tells us that the planet completes one rotation in that time. To determine the distance from the center of rotation to the equator (r), we need to consider that the circumference of a circle is given by the formula:

C = 2 * pi * r,

where:
- C is the circumference,
- pi is a mathematical constant approximately equal to 3.14159,
- r is the radius.

Given that the rotational period is 15 hours, we convert it to seconds (3600 seconds per hour) and find that the planet completes one rotation in 54,000 seconds.

To find the radius of the planet, we divide the circumference by 2 * pi:

r = C / (2 * pi) = (54,000 seconds * 2 * pi) / (2 * pi) = 54,000 seconds.

Now, we can calculate the apparent weight at the equator using the gravitational force formula:

F = (G * m1 * m2) / r^2,

where m1 is the astronaut's mass (90 kg), G is the gravitational constant, m2 is the mass of the planet (5.972 x 10^24 kg), and r is the radius of the planet (54,000 seconds).

Simplifying the equation, we have:

F = (6.67430 x 10^-11 N m^2/kg^2 * 90 kg * 5.972 x 10^24 kg) / (54,000 seconds)^2.

Evaluating this expression, we find the apparent weight of the astronaut at the equator of the planet.