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Circular Motion and Gravitation

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Circular Motion and Gravitation

Centuries ago, who would have thought that the same laws of mechanics that govern the motion of objects here on Earth also dictate the motion of the planets? This wasn't always obvious. While objects on Earth tend to fall in straight lines towards its center, the Moon moves in circles around the Earth. We owe our modern understanding that the same laws of mechanics governing the motion of objects here on Earth also apply to the Moon and planets to Isaac Newton. Indeed, Newton related circular motion to planetary movement when he discovered his law of universal gravitation. He had figured that there was a gravitational force that caused objects to fall toward the Earth since they accelerate on the way down. He then wondered what caused the Moon to orbit around the Earth and realized that gravity could extend beyond the surface of the Earth, even so far as to be the force that kept the Moon in a circular path. In this article, we will first discuss uniform circular motion and then relate the principles of circular motion to gravitation. We'll discuss circular motion and gravitation definitions, examples, formulas, and two worked-out questions.

Uniform Circular Motion

Circular motion refers to any kind of movement that follows a circular path, but there's a specific type of circular motion called uniform circular motion.

Uniform circular motion is the motion of an object moving with constant velocity in a circle with a fixed radius.

When an object moves in a uniform circular motion, as in the image below, its velocity always points in the direction tangent to the circle, as shown by the red arrow. This means the direction of the velocity is constantly changing as the object moves around the circle. The acceleration of the object points towards the center of the circle, shown by the yellow arrow, to account for the changing direction of the velocity. This acceleration is called centripetal acceleration.

Centripetal acceleration is the acceleration of an object moving in a circular motion.

Circular Motion and Gravitation Uniform Circular Motion StudySmarterA free-body diagram of uniform circular motion showing the directions of velocity and acceleration, StudySmarter Originals

Centripetal Force

We know from Newton's second law that there must be a force acting on an object if it has an acceleration. We call forces centripetal whenever they cause centripetal acceleration.

Centripetal force is a force that causes an object to follow a circular path.

The centripetal force has the same direction as the centripetal acceleration–towards the center of the circular path (the name "centripetal" literally means "center-seeking"). Centripetal force isn't a specific type of force; we use the term centripetal force to describe any force keeping an object in a circular motion. In a uniform circular motion, the centripetal force doesn't cause the object to move to the center of the circle because the object is moving so fast that it keeps spinning in a circle.

If you attached a ball to a string and swung it in a circle above your head at a constant speed, the ball would be in a uniform circular motion. The tension of the string is the centripetal force at any given moment. Hence, the net force in the plane of the circle is

\[F_{\text{net}} = F_c = F_T\]

The centripetal force causes the ball to change directions to continuously spin in a circle.

Imagine now that you let go of the string. At the instant you let go of the string, the net force in the plane of the circle becomes zero,

\[F_{\text{net}} = 0,\]

such that, by Newton's first law, the ball would continue moving with the velocity it had at that moment. As in the figure above, this would be a vector tangent to the circle at the point of release.

Gravitation and Circular Motion Definition

Circular motion is very relevant to gravitation, as many objects in space follow circular orbits due to gravitation.

Gravitation, or the gravitational force, is the force of attraction that all objects with mass exert on each other.

Since all massive objects experience the force of gravitation, we call it a fundamental force. Other fundamental forces are the electromagnetic force and the strong and weak nuclear forces.

The image below depicts the gravitational force. Two masses separated at a distance \(r\) exert a gravitational force on each other. This pulls them closer together. Every object with mass exerts a gravitational force on other objects, but most of the time an object has to have a large mass for us to be able to measure its gravitational pull. This is why we mainly discuss gravitation at the scale of planets, stars, and galaxies.

Circular Motion and Gravitation Definition StudySmarterTwo masses are attracted together by the gravitational force, Wikimedia Commons

Close to the Earth's surface, gravitation causes all objects to fall down. Since all objects experience the same force pointing in the same direction, we can locally represent the effect of gravity by means of a vector field.

Conversely, if we zoom out past the Earth's surface and consider the effect of its gravitational pull on artificial satellites and the Moon, we now find that the force of Earth's gravity causes these objects to orbit in a circular motion. For an object in orbit, the gravitational force acts as a centripetal force. The force is directed toward the massive object at the center of the circular path, causing the orbiting object's velocity to change direction to keep the object traveling in a circle. The orbiting object's velocity is high enough to keep the gravitational force from pulling it straight toward the massive object.

Kepler's great discovery was that the orbits of planets around the Sun are not exactly circular. Rather, they are ellipses. One number we use to characterize ellipses is their eccentricity, which is a measure of how much their shape deviates from that of a circle. More specifically, ellipses with an eccentricity \(e = 0\) are circles. As is the case, the eccentricity of Earth's orbit around the Sun is

\[e_{\text{E}} \approx 0.01671 \]

while the eccentricity of the Moon's orbit around the Earth is

\[e_{\text{M}} \approx 0.0549. \]

Since these values are nearly zero, we can treat the gravitational force in these cases as a uniform centripetal force to great accuracy.

Gravitation and Circular Motion Examples

Some examples of gravitation and circular motion include the following:

  • Earth's gravity pulls the Moon in an orbit with circular motion.
  • Satellites orbit the Earth at a calculated constant speed that allows them to have uniform circular motion.
  • All the planets in our solar system are pulled into circular motion by the Sun's gravity.

Let's take a further look at the example of the Moon orbiting the Earth, shown in the image below:

Circular Motion and Gravitation Moon Orbit Example StudySmarterThe Moon orbiting the Earth, adapted from images by Vecteezy

The gravitational force that the Earth exerts on the Moon acts as the centripetal force, directed toward the center of the Earth (shown by the blue arrow). This force creates a centripetal acceleration, also directed toward the center of the Earth, which causes the velocity of the Moon to consistently change direction in a circular pattern. The velocity of the Moon is always directed tangentially to the orbit or perpendicular to the centripetal force and acceleration (the velocity is shown by the red arrow). If the Earth's gravity were to suddenly stop, the Moon would go flying off into space straight from whatever point it is released from, just as in the case of the ball attached to the string we discussed above.

Gravitation and Circular Motion Formulas

Below we'll discuss some of the formulas that are most relevant to gravitation and circular motion.

Gravitation Formula

Newton's law of universal gravitation describes the equation for the gravitational force between two objects:

\[F_{\text{g}} = G \frac{m_1 m_2}{r^2}\]

where \(m_1\) and \(m_2\) are the masses of two objects, \(r\) is the distance between the centers of the masses, and \(G\) is the gravitational constant, which is \(G = 6.67 \times 10^{-11} \, \mathrm{m}^3 / (\mathrm{kg}\,\mathrm{s}^2)\). This formula is especially relevant at large distances; on a planet's surface we might use the equation

\[F_{\text{g}} = mg\]

to describe the gravitational force, but this equation is only accurate if the gravitational field is constant. At large distances, when the gravitational field isn't constant, we have to use the equation above to find the gravitational force.

Circular Motion - Velocity Formula

In circular motion, velocity is still equal to the change in distance over the change in time; the distance is just in a circular path instead of a straight one. So the formula for velocity, \(v\), of an object moving in a circle would be the circumference of the circle, \(2\pi r\), divided by the time it takes the object to complete one revolution, \(T\):

\[v = \frac{2\pi r}{T}.\]

Velocity is measured in meters-per-second (\(\mathrm{m}/\mathrm{s}\)), the radius is in meters (\(\mathrm{m}\)), and the time to complete a revolution is in seconds (\(\mathrm{s}\)).

Circular Motion - Centripetal Acceleration Formula

The formula for centripetal acceleration is as follows:

\[a_c = \frac{v^2}{r}.\]

\(v\) is the velocity of the object in meters-per-second (\(\mathrm{m}/\mathrm{s}\)) and \(r\) is the radius of the circular path of the object in meters (\(\mathrm{m}\)).

Circular Motion - Centripetal Force Formula

Since force is equal to mass times acceleration, we can multiply the centripetal acceleration formula above by mass to get the equation for centripetal force:

\[\begin{align} F_c &= ma_c\\ &= \frac{mv^2}{r}.\end{align}\]

\(F\) represents the centripetal force in newtons (\(\mathrm{N}\)), \(m\) is the mass of the orbiting object in kilograms (\(\mathrm{kg}\)), \(v\) is the velocity of the object in meters-per-second (\(\mathrm{m}/\mathrm{s}\)), and \(r\) is the radius of the object’s orbit in meters (\(\mathrm{m}\)).

Gravitation and Circular Motion Physics Questions

Below is an example of a gravitational and circular motion question and its solution.

A \(200 \, \mathrm{kg}\) satellite is in a circular orbit \(30\,000\,\mathrm{km}\) from the Earth's surface. What is the velocity of the satellite? Use \(6371\, \mathrm{km}\) for Earth's radius and \(5.98\times 10^{24} \, \mathrm{kg}\) for its mass.

The only force acting on the satellite is the gravitational force, so the gravitational force will equal the satellite's mass, \(m_\text{s}\), times its centripetal acceleration:

\[F_{\text{g}} = m_\text{s}a_c.\]

We need to use the universal gravitation law for the gravitational force since the satellite is far from the Earth's surface:

\[F_{\text{g}} = G \frac{m_\text{s} M}{r^2}.\]

Here, \(M\) denotes the mass of the Earth. We also have the equation for centripetal acceleration:

\[a_c = \frac{v^2}{r}.\]

Substituting these two equations into our first equation we get the following:

\[G \frac{m_\text{s} M}{r^2} = m_\text{s} \frac{v^2}{r}.\]

Simplifying and solving for \(v\):

\[v = \sqrt{\frac{GM}{r}}.\]

We can then plug in our given numbers. Before doing so, note that the value we need to use for \(r\) is not merely the distance of the satellite from the Earth. Rather, it's the sum of the Earth's radius and this distance. The reason we have to account for this is that Newton's law of universal gravitation requires the distance between the centers of the masses. Accounting for this, we have:

\[\begin{align} v &= \sqrt{\frac{(6.67 \times 10^{-11} \, \mathrm{m}^3 / (\mathrm{kg}\,\mathrm{s}^2))(5.98\times 10^{24} \, \mathrm{kg})}{(3 \times 10^7 \,\mathrm{m} + 6.371 \times 10^6 \, \mathrm{m})}} \\ &= 3\,275 \, \mathrm{m}/\mathrm{s}.\end{align}\]

The velocity of the satellite is \(3\,275 \, \mathrm{m}/\mathrm{s}\).

Let's consider a similar example with the main difference that, instead of orbiting the Earth, the satellite is now orbiting the Moon

A \(200 \, \mathrm{kg}\) satellite is in a circular orbit \(30\,000\,\mathrm{km}\) from the Moon's surface. What is the velocity of the satellite? Use \(1737\, \mathrm{km}\) for the Moon's radius and \(7.35\times 10^{22} \, \mathrm{kg}\) for its mass.

Since the situation is nearly identical to that of the previous example, we can proceed right away by using the equation we derived before for the satellite's velocity:

\[v = \sqrt{\frac{GM}{r}}.\]

Here, we have to be careful to use the correct values for the Moon instead of those of the Earth. Substituting the given values, we have:

\[\begin{align} v &= \sqrt{\frac{(6.67 \times 10^{-11} \, \mathrm{m}^3 / (\mathrm{kg}\,\mathrm{s}^2))(7.35\times 10^{22} \, \mathrm{kg})}{(3 \times 10^7 \,\mathrm{m} + 1.737 \times 10^6 \, \mathrm{m})}} \\ &= 393 \, \mathrm{m}/\mathrm{s}.\end{align}\]

In this case, the velocity of the satellite is \(393 \, \mathrm{m}/\mathrm{s}\).

Circular Motion and Gravitation - Key takeaways

  • Uniform circular motion is the motion of an object moving with constant velocity in a circle with a fixed radius.
  • Centripetal acceleration is the acceleration of an object moving in a circular motion, acting toward the center of the circle.
  • Centripetal force is a force that causes an object to follow a circular path. It acts toward the center of the circle.
  • Gravitation, or the gravitational force, is the force of attraction that all objects with mass exert on each other.
  • The gravitational force acts as the centripetal force for objects in orbit.
  • The equation for the gravitational force between two objects is \[F_{\text{g}} = G \frac{m_1 m_2}{r^2}.\]
  • The equation for velocity in circular motion is \[v = \frac{2\pi r}{T}.\]
  • The formula for centripetal acceleration is \[a_c = \frac{v^2}{r}.\] and the equation for centripetal force is \(F_c = \frac{mv^2}{r}\).

Frequently Asked Questions about Circular Motion and Gravitation

Gravitation is the force of attraction that two objects with mass exert on each other. Circular motion is movement in a circular path. Gravitation can cause circular motion by pulling objects (such as planets) into a circular orbit.

There isn't an overall formula for circular motion. The formulas relevant to circular motion include velocity (), centripetal acceleration () , and centripetal force () .

Yes, gravitational force affects circular motion. It is especially relevant to orbits in space--the gravitational force acts as the centripetal force that causes circular motion. It can also affect other forms of circular motion such as a roller coaster or pendulum.

The main force that affects circular motion is known as the centripetal force. The centripetal force is not a specific type of force and can be any force that causes an object to move in a circular path, such as tension or gravity.

Circular motion is caused by a centripetal force. It causes an object's velocity to consistently change directions into a circular path due to the force being directed toward the center of the circle.

Final Circular Motion and Gravitation Quiz

Question

What is the definition of Keplers third law?

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Answer

The law of periods - the orbital period squared is proportional to the distance between the bodies cubed.

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Question

What is the definition of Keplers second law?

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Answer

The law of equal areas - equal area is swept out in equal time, for a line connecting the sun and a planet.

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What is the definition of Kepler’s first law?

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Answer

The law of ellipses - planetary orbits are ellipses or other conic sections, with the sun at a fixed point.

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Question

True or false: an orbit can be hyperbolic in shape.

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Answer

True.

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Question

The change in a planet’s distance from a star is ... to the change in the planet’s orbital velocity.

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Answer

Inversely proportional.

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Question

Give an example of conservation of angular momentum.

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Answer

An ice or roller skater is rotating circularly with their arms held out. When the skater pulls their arms in close to their chest, their rotational velocity increases.

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Question

Give an example of one of Kepler’s laws used for a system other than the sun and planet.

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Answer

  • An artificial satellite orbiting Earth.
  • A comet with a highly elliptical orbit orbiting a star with a period of several hundred years.
  • Exoplanets orbiting another star outside our solar system obey the law of conservation of angular momentum.

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Question

True or false: Kepler’s laws are not approximations.

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Answer

False.

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Question

Why do orbital periods have no dependence on eccentricity?

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Answer

The closer an orbiting object is to a star, the faster it will travel, and vice versa. Kepler’s second law says that equal areas will be swept out in equal times. The changes in velocity for a planet with an elliptical orbit result in the same period if it had a circular orbit. Conservation of angular momentum keeps this equal, regardless of how elongated an orbit is.

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Question

What does Newton’s law of universal gravitation state?

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Answer

Any two masses in the universe will experience the attractive force of gravity, regardless of size.

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True or false: orbiting satellites are in free fall.

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Answer

True.

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Question

What is a gravitational field?

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Answer

A vector field that models the gravitational force exerted on a small unit of mass, anywhere around the central mass of the field.

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Question

What is the reason for the sensation of apparent “weightlessness” in space?

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Answer

A lack of contact forces during free fall, where the force of gravity cannot be felt directly.

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True or false: both your mass and your weight will change when measured on planets with different gravities.

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Answer

False.

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True or false: the acceleration due to gravity is the same anywhere on Earth.

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Answer

False.

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Fill in the blanks: Mass is a ... quantity and weight is a ... quantity.

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Scalar, vector.

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What is a fundamental force?

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Answer

An interaction that drives the physical processes and composition of the universe, and cannot be broken down into any more basic processes.

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What is the weakest fundamental force?

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Gravity.

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Question

What is a vector field, and why are they useful for describing forces?

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  • A vector field visualizes the direction and magnitude of a force vector at many different points in space.
  • We can use vector fields to model the force exerted on a small mass at any point in space within the field.

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Question

What is the gravitational interaction?

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The universal attraction of all masses to one another.

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What is the electromagnetic interaction?

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The attracting and repelling interactions between particles with charges.

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What is the weak nuclear interaction?

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The interaction that drives certain radioactive and decay processes between electrons, protons, and neutrons in atoms.

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What is the strong nuclear interaction?

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The interaction that binds atomic nuclei, as well as particles like protons and neutrons, together.

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Question

Which force dominates on an astronomical scale?

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Answer

Gravitational force.

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Give an example of electromagnetic interaction.

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  • Auroras occur from charged particles spinning through a planet's magnetic field.
  • Speakers use magnetic coils and convert electrical energy into sound.
  • Static electricity builds up on clothing in a dryer machine due to charges transferred between materials; the static shock is a discharge of the extra electrons.

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Give an example of the weak nuclear interaction.

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Answer

  • Energy and new particles are emitted during atomic fusion.
  • Beta-decay of the radioactive isotope carbon-14 in organic material allows us to carbon date organic material.

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True or false: the electric repulsion of protons in an atomic nucleus dominates over the strong nuclear force.

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Answer

False.

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What is nuclear fusion?

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Two atoms fuse together to create a heavier atom due to the strong force attracting the nuclei together. An additional particle and a large amount of energy are emitted in the process due to the weak interaction.

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Why is the spatial scale we study physical phenomena important?

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Answer

The spatial scale changes which force dominates.

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Contact forces, such as the normal force and frictional force, are examples of which fundamental interaction?

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Answer

Electromagnetic.

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Why do we consider electric and magnetic forces to be one fundamental interaction?

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Answer

Both are forces due to interactions between charged particles.

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What type of law does the distance for gravitational and electric forces obey?

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Answer

Inverse square distance.

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What is the SI unit for charge?

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Coulomb.

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True or false: the gravitational force can be both attractive and repulsive.

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Answer

False.

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True or false: the electric force can be both attractive and repulsive.


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Answer

True.

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Question

If we increase the distance between two charges by a factor of 4, the electric force is ... by a factor of ...

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Answer

Decreased, 16.

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If we decrease the distance between two masses by a factor of 2, the gravitational force is ... by a factor of ... .

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Answer

Increased, 4.

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How does the force of gravity relate to mass?

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Proportional to the product of the masses.

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True or false: the gravitational force from a satellite orbiting the Earth is equal in magnitude to the force from the Earth on the satellite.

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Answer

True.

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Question

What is the distance range for gravitational and electric forces?

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Answer

Infinite.

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Question

Give an example of the electric force.

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Answer

  • The static shock of touching a doorknob after walking across a carpet.
  • After rubbing a balloon with your shirt, your hair sticks up with contact to the balloon.

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The electrostatic force means that charges are ... .

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Answer

At rest.

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How does classical physics describe the theory of gravity?

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Answer

Gravity is a universal attracting force between all masses at infinite distances, regardless of size.

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What is gravitational mass?

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The mass of an object measured under the effects of gravity.

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What is inertial mass?

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The mass of an object as measured by a weighing scale.

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Why aren’t weighing scales an instrument we can measure mass with?

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Answer

A weighing scale measures the force of a mass accelerating downwards due to gravity, with units in Newtons or pound-force.

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Which law states that inertial mass and gravitational mass are identical?

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Answer

The equivalence principle.

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Which of the following are examples of inertial mass?

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Answer

A textbook resting on a desk will not move until a force is applied.

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How does inertial mass relate to an object’s resistance to motion? Give an example of this relationship.

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  • An object with a greater inertial mass will need a greater force applied to move it from rest; more massive objects are more resistant of a change in motion.
  • Pushing a book across a table with a smooth, polished surface requires less force than pushing a large couch across a carpeted floor.

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Question

How does the force of friction relate to the measurement of inertial mass?

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Answer

Increased friction between two surfaces requires a greater applied force to move the same mass, so the force of friction should be included in the net force.

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