Uniform Circular Motion

What Coefficient of Friction Do Car Tires Need on a Flat Curve?

(a) Calculate the centripetal force exerted on a 900 kg car that negotiates a 500 m radius curve at 25.0 m/s.

Centripetal Acceleration Expressions

By using the expressions for centripetal acceleration 𝑎𝑐 from 𝑎𝑐=𝑣2/𝑟; 𝑎𝑐=𝑟𝜔², we get two expressions for the centripetal force Fc in terms of mass, velocity, angular velocity, and radius of curvature:

Kepler's Second Law

Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times

banked curves

Let us now consider banked curves, where the slope of the road helps you negotiate the curve. See Figure 6.13. The greater the angle 𝜃, the faster you can take the curve. Race tracks for bikes as well as cars, for example, often have steeply banked curves. In an "ideally banked curve," the angle 𝜃 is such that you can negotiate the curve at a certain speed without the aid of friction between the tires and the road. We will derive an expression for 𝜃 for an ideally banked curve and consider an example related to it.

Centripetal Force

Notice that the object moving in this example is experiencing a net force that always points towards the center of the resulting circular motion. This type of net force gets a special name: centripetal force. Centripetal is Latin for "center-seeking." In other words, the net force in this example is a "center-seeking" net force.

Tides

Ocean tides are one very observable result of the Moon's gravity acting on Earth. Figure 6.24 is a simplified drawing of the Moon's position relative to the tides. Because water easily flows on Earth's surface, a high tide is created on the side of Earth nearest to the Moon, where the Moon's gravitational pull is strongest. Why is there also a high tide on the opposite side of Earth? The answer is that Earth is pulled toward the Moon more than the water on the far side, because Earth is closer to the Moon. So the water on the side of Earth closest to the Moon is pulled away from Earth, and Earth is pulled away from water on the far side. As Earth rotates, the tidal bulge (an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits) keeps its orientation with the Moon. Thus there are two tides per day (the actual tidal period is about 12 hours and 25.2 minutes), because the Moon moves in its orbit each day as well).

Explain why the maximum possible speed of a car turning a corner will be larger if the road is banked with a large angle.

The force vector diagram for the car would look like:

Sections

⚛️ 6.1 Centripetal Force ⚛️ 6.2 Universal Law of Gravity ⚛️ 6.3 Circular Orbits

Centripetal force problem solving

⚛️ Draw Diagram ⚛️ Apply equations

All these motions are governed by gravitational force, and it is possible to describe them to various degrees of precision. Precise descriptions of complex systems must be made with large computers. However, we can describe an important class of orbits without the use of computers, and we shall find it instructive to study them. These orbits have the following characteristics:

1. A small mass 𝑚 orbits a much larger mass 𝑀. This allows us to view the motion as if 𝑀were stationary—in fact, as if from an inertial frame of reference placed on 𝑀 —without significant error. Mass 𝑚 is the satellite of 𝑀, if the orbit is gravitationally bound. 2.The system is isolated from other masses. This allows us to neglect any small effects due to outside masses.

Forces: centripetal or radial acceleration

Any force or combination of forces can cause a centripetal or radial acceleration. Just a few examples are the tension in the rope on a tether ball, the force of Earth's gravity on the Moon, friction between roller skates and a rink floor, a banked roadway's force on a car, and forces on the tube of a spinning centrifuge.

A 750 kg car travels with a constant speed of 8 m/s over a hill with a radius of curvature of 15.0 m. What is the normal force acting on the car when it has reached the top of the hill?

As usual, start with a force vector diagram. When the car is at the top of the hill, the center of the circle is down, which means the centripetal (net) force is also down. This means that the force of earth (gravity) on the car must be larger than the force of the ground (normal force) on the car. Also, remember that the centripetal acceleration is also down (negative).

What Is the Ideal Speed to Take a Steeply Banked Tight Curve?

Curves on some test tracks and race courses, such as the Daytona International Speedway in Florida, are very steeply banked. This banking, with the aid of tire friction and very stable car configurations, allows the curves to be taken at very high speed. To illustrate, calculate the speed at which a 100 m radius curve banked at 65.0° should be driven if the road is frictionless.

Outer Space

Examples of gravitational orbits abound. Hundreds of artificial satellites orbit Earth together with thousands of pieces of debris. The Moon's orbit about Earth has intrigued humans from time immemorial. The orbits of planets, asteroids, meteors, and comets about the Sun are no less interesting. If we look further, we see almost unimaginable numbers of stars, galaxies, and other celestial objects orbiting one another and interacting through gravity.

ideal banking

For ideal banking, the net external force equals the horizontal centripetal force in the absence of friction. The components of the normal force N in the horizontal and vertical directions must equal the centripetal force and the weight of the car, respectively. In cases in which forces are not parallel, it is most convenient to consider components along perpendicular axes—in this case, the vertical and horizontal directions.

The distance from the Earth to the sun is about 149.6 million km, and the distance from the sun to Jupiter is about 778.5 million km. It takes the Earth 365.25 days to orbit the sun. What is the approximate length of one year on Jupiter?

Since this is a ratio, as long as both sides of the equation use the same units, you don't need to convert the units to SI values.

What is the velocity of the international space station (ISS)? It orbits about 400 km above the surface of the Earth. The Earth has a mass of approximately 5.97x10²⁴ kg and a radius of about 6371 km. How much time will it take the ISS to orbit the earth once?

The ISS orbits the Earth once roughly every hour and a half. That means the astronauts also get to experience a "sunrise" once every hour and a half. Also, this is the time it would take any object 400 km from the Earth to orbit once.

Sun Tides

The Sun also affects tides, although it has about half the effect of the Moon. However, the largest tides, called spring tides, occur when Earth, the Moon, and the Sun are aligned. The smallest tides, called neap tides, occur when the Sun is at a 90º size angle to the Earth-Moon alignment.

Kepler and Brahe

The conditions are satisfied, to good approximation, by Earth's satellites (including the Moon), by objects orbiting the Sun, and by the satellites of other planets. Historically, planets were studied first, and there is a classical set of three laws, called Kepler's laws of planetary motion, that describe the orbits of all bodies satisfying the two previous conditions (not just planets in our solar system). These descriptive laws are named for the German astronomer Johannes Kepler (1571-1630), who devised them after careful study (over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe (1546-1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute the evidence from which new interpretations and meanings can be constructed.

An engineer wants to determine the speed limit to post on a particular hill. The hill has a radius of curvature of 15 m. What is the maximum speed that a car can travel on the hill and still stay on the road (i.e., no projectile motion)?

The larger the speed, the larger the centripetal acceleration. Large centripetal accelerations are the result of large net forces. This means the centripetal force on the car needs to be maximized. Draw a force diagram for the forces on the car when it is at the top of the hill:

MISCONCEPTION ALERT

The magnitude of the force on each object (one has larger mass than the other) is the same, consistent with Newton's third law.

Kepler's First Law

The orbit of each planet about the Sun is an ellipse with the Sun at one focus.

Kepler's Third Law

The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun. In equation form, this is where 𝑇 is the period (time for one orbit) and 𝑟 is the average radius. This equation is valid only for comparing two small masses orbiting the same large one. Most importantly, this is a descriptive equation only, giving no information as to the cause of the equality.

A student spins a toy attached to a 0.4 m string around their head parallel to the floor. The toy rotates once every 0.6 seconds. What is the toy's acceleration?

The time for an object to complete one rotation is called the object's period, which uses the symbol T. Since this is still a time value, the units are seconds.

Why does Earth not remain stationary as the Moon orbits it?

This is because, as expected from Newton's third law, if Earth exerts a force on the Moon, then the Moon should exert an equal and opposite force on Earth (see Figure 6.23). We do not sense the Moon's effect on Earth's motion, because the Moon's gravity moves our bodies right along with Earth but there are other signs on Earth that clearly show the effect of the Moon's gravitational force as discussed in Satellites and Kepler's Laws: An Argument for Simplicity.

Black Hole

Tides are not unique to Earth but occur in many astronomical systems. The most extreme tides occur where the gravitational force is the strongest and varies most rapidly, such as near black holes (see Figure 6.26). A few likely candidates for black holes have been observed in our galaxy. These have masses greater than the Sun but have diameters only a few kilometers across. The tidal forces near them are so great that they can actually tear matter from a companion star.

The only variables that affect the orbital speed are the mass of the planet and the distance from the center of the planet. Objects A and B will have the largest speed since they are closest to the planet and so experience a larger falling acceleration than C. The masses of the objects do not affect their overall falling motion.

VA = VB > VC

Newton's Universal Law of Gravitation

What do aching feet, a falling apple, and the orbit of the Moon have in common? Each is caused by the gravitational force. Our feet are strained by supporting our weight—the force of Earth's gravity on us. An apple falls from a tree because of the same force acting a few meters above Earth's surface. And the Moon orbits Earth because gravity is able to supply the necessary centripetal force at a distance of hundreds of millions of meters. In fact, the same force causes planets to orbit the Sun, stars to orbit the center of the galaxy, and galaxies to cluster together. Gravity is another example of underlying simplicity in nature. It is the weakest of the four basic forces found in nature, and in some ways the least understood. It is a force that acts at a distance, without physical contact, and is expressed by a formula that is valid everywhere in the universe, for masses and distances that vary from the tiny to the immense.

Uniform Circular Motion and Gravitation

⚛️ 6.2 Centripetal Acceleration ⚛️ 6.3. Centripetal Force ⚛️ 6.5. Newton's Universal Law of Gravitation ⚛️ 6.6 Satellites and Kepler's Laws: An Argument for Simplicity

centrifuge

⚛️ A centrifuge (see Figure 6.9b) is a rotating device used to separate specimens of different densities. ⚛️ High centripetal acceleration significantly decreases the time it takes for separation to occur, and makes separation possible with small samples. ⚛️ Centrifuges are used in a variety of applications in science and medicine, including the separation of single cell suspensions such as bacteria, viruses, and blood cells from a liquid medium and the separation of macromolecules, such as DNA and protein, from a solution. ⚛️ Centrifuges are often rated in terms of their centripetal acceleration relative to acceleration due to gravity (𝑔) maximum centripetal acceleration of several hundred thousand 𝑔 is possible in a vacuum. ⚛️ Human centrifuges, extremely large centrifuges, have been used to test the tolerance of astronauts to the effects of accelerations larger than that of Earth's gravity.

Centripetal Acceleration!

⚛️ According to Newton's second law, a net force must always result in an acceleration. However, the velocity vector arrows in this example are always the same size. This means that the speed of the rotating object is constant. How can the object have a constant speed AND an acceleration? ⚛️ Be careful here - remember that acceleration is the rate at which the velocity changes with time. Velocity is a vector measurement that includes speed AND direction. Even though the rotating object keeps a constant speed, it does NOT keep a constant direction. Since the direction changes throughout the rotation, the object must experience a change in velocity. If the object's velocity changes, the the object must also experience an acceleration. ⚛️ The last lesson showed how acceleration can have a constant direction motion with a changing speed. This lesson shows that an object can have a constant speed motion while changing direction. BOTH of these examples require the object to experience an acceleration. ⚛️ The next question is: what is the direction of this acceleration? Recall that Lesson 5 showed that an object's acceleration is always in the same direction as the net force acting on the object. Since the net force in this example is center-seeking, the acceleration will also be center-seeking and is thus called centripetal acceleration. ⚛️ Where is the centripetal acceleration, v is the velocity, and r is the radius of the circular path (usually called the radius of curvature).

centripetal force

⚛️ Any net force causing uniform circular motion is called a centripetal force. ⚛️ The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. ⚛️ According to Newton's second law of motion, net force is mass times acceleration: net F=𝑚𝑎. ⚛️ For uniform circular motion, the acceleration is the centripetal acceleration— 𝑎=𝑎𝑐 ⚛️ Thus, the magnitude of centripetal force Fc

Unknown Gravitational Field Strength.

⚛️ As shown in the last example, both methods of calculating the weight of an object yield virtually the same results (depending on the precision of the measurements plugged into the equations). ⚛️ This equation can also be used to solve for an unknown gravitational field strength.

centripetal force :)

⚛️ As usual, net forces cause accelerations. ⚛️ When the net force causes centripetal acceleration, it is called centripetal force. ⚛️ The centripetal force points towards the center of the circular path.

Gravity Forces vs Gravitational Strength

⚛️ Be very careful; it is common for students to confuse gravity forces with gravitational field strength. ⚛️ The force is the amount of push or pull between two masses. ⚛️ The field strength is the ability of a single mass to push or pull theoretical masses at various distances from that source mass. ⚛️ Be sure to read the questions carefully in order to solve for the correct variable.

Net Force and Centripetal Acceleration

⚛️ Centripetal acceleration can also be used to calculate the net force acting on the object. Newton's second law states that: ⚛️ ∑F = ma Replace ac with and then: ⚛️ Where ΣF is the sum of all forces in one direction (centripetal force), m is the mass, v is the velocity, and r is the radius of curvature. ⚛️ Some textbooks use the symbol ∑Fc or just to represent the centripetal force.

centripetal force 1

⚛️ Consider an object moving to the right at a constant speed. The object experiences a brief downward perpendicular force. ⚛️ No horizontal forces are present, so the object will still move to the right but with an added downward component:

Lesson Objectives

⚛️ Define and calculate the magnitude and direction of centripetal acceleration ⚛️ Define and calculate the magnitude and direction of centripetal force ⚛️ Use Newton's law of gravity to calculate the gravitational force between two or more objects ⚛️ Define Kepler's three laws of planetary motion

gravitational force

⚛️ The gravitational force acts as the centripetal force for objects in orbit. ⚛️ Orbital motion is essentially freefall. ⚛️ The velocity of a circular orbit can be determined by combining Newton's law of gravity with the definition of centripetal acceleration.

A student attaches a rope to a toy and uses the rope to rotate the toy in a circle on a countertop. The picture below shows a top-down view of the motion. When the toy reaches point P, the student releases the rope. Which of the following arrows correctly indicates the path the toy will take? (Assume negligible friction.)

⚛️ Draw a force diagram for the forces on the toy. The normal force of the counter and the pull of the Earth cancel (Newton's first law). The only force acting in the horizontal direction is the tension force of the rope on the toy. The net force (i.e., centripetal force) is equal to the tension force. Note that the tension force is always perpendicular to the direction of the motion of the toy at any given point causing the toy to move in a circle. ⚛️ Once the tension force is removed, no horizontal forces will act on the toy. According to Newton's first law, this means that the object must experience constant velocity. The object will continue to move in whatever direction and speed the toy had at the point it is released. ⚛️ The correct answer is B.

Figure

⚛️ Figure 6.8 shows an object moving in a circular path at constant speed. The direction of the instantaneous velocity is shown at two points along the path. ⚛️ Acceleration is in the direction of the change in velocity, which points directly toward the center of rotation (the center of the circular path). This pointing is shown with the vector diagram in the figure. We call the acceleration of an object moving in uniform circular motion (resulting from a net external force) the centripetal acceleration. (𝑎c; centripetal means "toward the center" or "center seeking."

Universal Law of Gravity

⚛️ It's common to think of gravity as a force between some object and a planet or other celestial body (i.e., the weight of the object on that planet). However, the reality is that all objects with mass are gravitationally attracted to all other objects with mass. ⚛️ A student working at a desk is gravitationally attracted to their desk, their computer, the chair their sitting on, and everything else in their surroundings. ⚛️ Of course, according to Newton's third law, this also means that the desk, computer, chair, and everything else in the student's surroundings are also attracted to the student with an equal amount of force. ⚛️ Isaac Newton was able to determine the relationship between this force of attraction and the masses of two spherical objects:

Lab Objectives

⚛️ Lab: Calculate rotational speed ⚛️ Lab: Calculate centripetal force ⚛️ Lab: Accurately discuss the relationship between rotational speed and centripetal force

Newton's Law of Gravity

⚛️ Newton's law of gravity states that the any two masses will exert a gravitational force on one another. ⚛️ The gravitational force is proportional to both masses and the inverse square of the distance between the masses.

centripetal force 3

⚛️ Once again, a same-size perpendicular force can be applied to the new velocity. This new force will be directed to the left. ⚛️ The object will still have an unchanged downward velocity component but now with an added left component.

universal gravitational constant

⚛️ Since F has an inverse square relationship with r, the larger the distance between two objects, the smaller the gravity force of attraction becomes. This is not particularly surprising. However, it's interesting to note that F never reaches zero; as the distance between the two objects becomes infinitely large, the force approaches but never reaches zero. This means that all objects in the universe apply a non-zero gravity force on every other object in the universe. All objects experience a non-zero gravity force by every star, planet, etc., in the universe. Alternatively, all objects also applies a non-zero gravity force on every star, planet, etc., in the universe. ⚛️ Of course, this force is generally small enough that it might as well be zero. ⚛️ This equation could be used to calculate the weight of an object on or near the Earth.

Gravitational Orbits

⚛️ Some of Newton's contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. ⚛️ This theoretical prediction was a major triumph—it had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose a mechanism that caused them to follow these paths and not others.

center of mass

⚛️ The bodies we are dealing with tend to be large. To simplify the situation we assume that the body acts as if its entire mass is concentrated at one specific point called the center of mass (CM), which will be further explored in Linear Momentum and Collisions. ⚛️ For two bodies having masses 𝑚 and 𝑀 with a distance 𝑟 between their centers of mass, the equation for Newton's universal law of gravitation is

Keplers 3rd Law

⚛️ The circular path of an orbit is related to its period according to Kepler's third law ⚛️ Kepler's first law notes that the Earth's path around the sun is an ellipse with the sun at one of the foci. ⚛️ The second law states that planets move such that a line drawn between the planet and the sun will create equal areas in equal amounts of time.

The Case for Simplicity

⚛️ The development of the universal law of gravitation by Newton played a pivotal role in the history of ideas. While it is beyond the scope of this text to cover that history in any detail, we note some important points. The definition of planet set in 2006 by the International Astronomical Union (IAU) states that in the solar system, a planet is a celestial body that: 1) is in orbit around the Sun, 2) has sufficient mass to assume hydrostatic equilibrium and 3) has cleared the neighborhood around its orbit. ⚛️ A non-satellite body fulfilling only the first two of the above criteria is classified as "dwarf planet." In 2006, Pluto was demoted to a 'dwarf planet' after scientists revised their definition of what constitutes a "true" planet.

Centripetal Acceleration Equation

⚛️ The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? ⚛️ Note that the triangle formed by the velocity vectors and the one formed by the radii 𝑟and Δ𝑠 are similar. ⚛️ Both the triangles ABC and PQR are isosceles triangles (two equal sides). ⚛️ The two equal sides of the velocity vector triangle are the speeds 𝑣1=𝑣2=𝑣 ⚛️ Using the properties of two similar triangles, we obtain...

Newton's universal law of gravitation

⚛️ The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. ⚛️ Stated in modern language, Newton's universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them. ⚛️ The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Kepler's 3rd Law

⚛️ The most useful part of this equation, however, is that the values on the right side of the equation are all constants. You could use this to relate the periods of two objects orbiting the same planet or star without knowing anything about the planet or star itself. ⚛️ Note that for elliptical orbits, the ratio still holds true. However, the collection of constants to which the ratios are equal will be different.

Circular Orbits

⚛️ The next question that you might want to ask is: What is the speed that will allow the cannonball to achieve orbit rather than crashing into the ground? The exact velocity depends on the distance from the Earth. You can determine this relationship by using Newton's universal law of gravity along with Newton's second law and a bit of centripetal acceleration. ⚛️ The orbital velocity depends only on the mass of the planet and the distance from the center of the planet. The mass of the object does not matter. This relationship shouldn't be surprising since you learned in Lesson 2 that the mass of an object does not affect its falling acceleration. Also, the further away from the center of Earth the cannonball is, the smaller the orbital speed is. This relationship also makes sense since the Earth's gravitational field strength gets weaker at positions further from the Earth. The gravitational field strength is what determines an object's acceleration due to gravity.

Copernican Model

⚛️ The universal law of gravitation is a good example of a physical principle that is very broadly applicable. That single equation for the gravitational force describes all situations in which gravity acts. It gives a cause for a vast number of effects, such as the orbits of the planets and moons in the solar system. It epitomizes the underlying unity and simplicity of physics. ⚛️ Before the discoveries of Kepler, Copernicus, Galileo, Newton, and others, the solar system was thought to revolve around Earth as shown in Figure ⚛️ 6.31(a). This is called the Ptolemaic view, for the Greek philosopher who lived in the second century AD. This model is characterized by a list of facts for the motions of planets with no cause and effect explanation. There tended to be a different rule for each heavenly body and a general lack of simplicity. ⚛️ Figure 6.31(b) represents the modern or Copernican model. In this model, a small set of rules and a single underlying force explain not only all motions in the solar system, but all other situations involving gravity. The breadth and simplicity of the laws of physics are compelling. As our knowledge of nature has grown, the basic simplicity of its laws has become ever more evident.

centripetal force 4

⚛️ This can be take to its logical conclusion: ⚛️ This diagram shows the motion if brief time intervals pass between each application of the perpendicular force. What if the force is constantly applied with no time intervals? The result would be circular motion.

Centripetal force problem solving 2

⚛️ Time ⚛️ Position ⚛️ Velocity ⚛️ Acceleration

Centripetal Acceleration

⚛️ We know from kinematics that acceleration is a change in velocity, either in its magnitude or in its direction, or both. ⚛️ In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the magnitude of the velocity might be constant. ⚛️ You experience this acceleration yourself when you turn a corner in your car. (If you hold the wheel steady during a turn and move at constant speed, you are in uniform circular motion.) ⚛️ What you notice is a sideways acceleration because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration will become. ⚛️ In this section we examine the direction and magnitude of that acceleration.

centripetal force 2

⚛️ What if a force of the same magnitude is now added perpendicular to the new velocity? ⚛️ This new force will be directed down and to the left. The left component of the force will cancel the right component of the velocity leaving only a downward component.

What is the gravitational field strength near the surface of the planet Mercury? Mercury has a mass of 3.29x10²³ kg and a radius of 2440 km.

⚛️ What's interesting about this equation for gravitational field strength is that it shows that any object that has mass also has a gravitational field. ⚛️ This equation also demonstrates that the strength of a gravity field is proportional to the inverse square of the distance from the mass source, just like the gravity force.

centripetal acceleration :)

⚛️ When an object moves with uniform circular motion, the object moves in a circular path with a constant speed. ⚛️ Since the direction of the motion constantly changes, the velocity is not constant. ⚛️ Since a change in velocity exists, the object also experiences acceleration. The acceleration of an object moving with uniform circular motion is called centripetal acceleration and it points towards the center of the circular path.

Centripetal force 𝐹c

⚛️ You may use whichever expression for centripetal force is more convenient. ⚛️ Centripetal force 𝐹c is always perpendicular to the path and pointing to the center of curvature, because 𝐚𝑐 is perpendicular to the velocity and pointing to the center of curvature.

gravitational constant

⚛️ where 𝐹 is the magnitude of the gravitational force and 𝐺 is a proportionality factor called the gravitational constant. ⚛️ 𝐺 is a universal gravitational constant—that is, it is thought to be the same everywhere in the universe. It has been measured experimentally to be