A planet orbits a star along a circular path. The only force acting on that planet is the star’s gravitational force. Which of the following is true of the acceleration of the planet? (a) It has zero magnitude. (b) Neither its magnitude nor its direction is constant. (c) It is constant in direction but not in magnitude. (d) Its magnitude and direction are both constant. (e) It is constant in magnitude but not in direction.
Let’s begin by drawing our planet orbiting around our star in a circular path. The large yellow circle represents our star, S, and the small pink circle represents our planet, P. The dotted pink circle represents the circular path along which the planet orbits the star. We’re told that the only force acting on the planet is the star’s gravitational force. We use 𝐹 subscript g to represent the gravitational force.
Our problem wants to know what is true of the acceleration of the planet. We use the letter 𝑎 to represent acceleration. We now have two variables, force and acceleration, that we’re trying to find a relationship between. This should remind us of Newton’s second law. Newton’s second law tells us that the force on an object is equal to the mass of the object times the acceleration of the object.
We can apply Newton’s second law to our planet, replacing the force with the force of gravity. The mass refers to the mass of the planet. And we’re trying to determine what’s happening to the planet’s acceleration. While the planet orbits the star, its mass stays constant. It does not gain or lose anything. This tells us that the force of gravity is proportional to the acceleration.
If we want to determine what happens to the acceleration, then we must figure out what happens to the gravitational force on the planet as it orbits the star. To do this, we can use Newton’s universal law of gravitation. This law states that the force of gravitational attraction between two masses, 𝐹 g, is equal to the gravitational constant, 𝐺, times the mass of object one, 𝑚 one, times the mass of object two, 𝑚 two, divided by the distance between the objects squared, 𝑟 squared.
As our planet orbits the star, the gravitational constant, 𝐺, does not change. The mass of the planet, 𝑚 one, does not change. The mass of the star, 𝑚 two, does not change. And the distance between the center of the star and the center of the planet does not change because it’s traveling in a circular path, which means that it has a constant radius. All the variables that the force of gravity is dependent upon stay constant. This means that the force of gravity on the planet as it circles the star is also constant.
Because the force of gravity is proportional to the acceleration, we can say that the acceleration of the planet as it orbits the star is also constant. This means that we can eliminate any of our answer choices that do not have constant acceleration. We can see that there’ll be answer choice (b), neither its magnitude nor its direction is constant, and answer choice (c) it is constant in direction but not in magnitude.
We can also eliminate answer choice (a), it has zero magnitude, as we have seen that the force of gravity is dependent upon variables that are all nonzero. And since the mass of our planet is also nonzero, our acceleration, which is proportional to the force of gravity, must also be nonzero. This leaves us with answer choices (d) and (e). In order to determine which of our choices are correct, we must analyze the direction of the acceleration for our planet as it orbits the star.
Let’s return to our diagram and draw in the acceleration of our planet at different points along its circular path. Recall that when an object is in circular motion, it has a centripetal acceleration towards the center of the circle. We have drawn the acceleration of the planet at the position shown with a blue arrow pointing towards the center of the star. If we were to draw the acceleration of planet P at its new position, it would once again be directed towards the center of the star.
Even though these two acceleration arrows are pointing towards the center of the star, we can see that when the planet is in different positions, our acceleration is going to be in different directions. When our planet was in the first position, we can see that the acceleration was directed down and to the left of our screen as we look at it. Whereas when our planet is in its second position, we can see that the acceleration is directed down and to the right of our screen as we look at it. This tells us that the direction of our acceleration is not constant.
Now that we know that the direction of the acceleration is not constant, we can determine whether (d) or (e) is the correct answer choice. Our final answer in reference to the acceleration of the planet is (e). It is constant in magnitude but not in direction.