Rotational Motion and The Law of Gravity Ch 7

Rotation and Revolution Two types of circular motion are rotation and revolution. An axis is the straight line around which rotation takes place. When an object turns about an internal axis—that is, an axis located within the body of the object—the motion is called rotation, or spin. When an object turns about an external axis, the motion is called revolution.

The Ferris wheel turns about an axis. The Ferris wheel rotates, while the riders revolve about its axis. Earth undergoes both types of rotational motion. It revolves around the sun once every 365 ¼ days. It rotates around an axis passing through its geographical poles once every 24 hours.

Rotational Motion Solid objects undergo rotational motion A point on a rotating object undergoes circular motion Circular Motion is described in terms of the angle through which the point on the object moves

Centripetal Acceleration Tangential speed (v t ) depends on distance When tangential speed is constant, motion is described as uniform circular motion

An object moving in a circle at a constant speed still has an acceleration due to its change in direction Velocity is a vector so acceleration can be produced by a change in magnitude and direction Centripetal Acceleration is acceleration caused by a change in direction, directed toward the center of a circular path a c = Vt 2 / r

Centripetal Acceleration

a t and a c Are perpendicular and not the same thing a t is due to changing speed a c is due to change in direction To find the total (at & ac) use Pythagorean theorem Direction of total acceleration can be found using trig functions

Board Work A test car moves at a constant speed around a circular track. If the car is 48.2 m from the track’s center and has a centripetal acceleration of 8.05 m/s 2, what is the car’s tangential speed? The tub of a washing machine has a radius of 34 cm. During the spin cycle, the wall of tub rotates with a tangential speed of 5.5 m/s. Calculate the centripetal acceleration of the clothes against the tub

Causes of Circular Motion Centripetal Force: force that maintains circular motion This force is necessary for circular motion Ball moving in a circle: Δv due to Δ in direction a c is inward: a c = v t 2 /r Fc is used to change an objects straight line inertia F c = ma c = mv t 2 / r

Without a centripetal force, an object in motion continues along a straight-line path. With a centripetal force, an object in motion will be accelerated and change its direction.

Ff and Fc Inertia is often misinterpreted as a force Fc is the force directed toward the center and is necessary for circular motion Many times Fc is the force provided by friction If the force is lost the object leaves at a tangent to the circular motion

Force that maintains circular motion A pilot is flying a small plane at 56.6 m/s in a circular path with a radius of m. If a force of 18,900 N is needed to maintain the pilot’s circular motion, what is the plane’s mass? A kg car rounds a circular turn of radius 20.0 m If the road is flat and the coefficient of static friction between the tires and the road is 0.70, how fast can the car go without skidding?

Gravitational Force Orbiting objects are in free fall When objects are orbiting, the gravitational force between the object and Earth is a centripetal force that keeps the object in orbit Each successive cannonball has a greater initial speed, so the horizontal distance that the ball travels increases. If the initial speed is great enough, the curvature of Earth will cause the cannonball to continue falling without ever landing.

Newton’s Hypothesis Newton compared motion of the moon to a cannonball fired from the top of a high mountain. If a cannonball were fired with a small horizontal speed, it would follow a parabolic path and soon hit Earth below. Fired faster, its path would be less curved and it would hit Earth farther away. If the cannonball were fired fast enough, its path would become a circle and the cannonball would circle indefinitely.

This original drawing by Isaac Newton shows how a projectile fired fast enough would fall around Earth and become an Earth satellite.

Both the orbiting cannonball and the moon have a component of velocity parallel to Earth’s surface. This sideways or tangential velocity is sufficient to ensure nearly circular motion around Earth rather than into it. With no resistance to reduce its speed, the moon will continue “falling” around and around Earth indefinitely.

The gravitational force attracts Earth and the moon to each other. According to Newton’s 3 rd Law.

Newton’s Law of Gravitation Gravitational force: the mutual force of attraction between the particles of matter Keeps the planets orbiting around the sun Exists between any two masses regardless of size or composition Fg is inversely proportional to distance Distance increases, gravity decreases

Newton’s Law of Gravitation Fg is localized to the center of a spherical mass Fg = G (m 1 m 2 /r 2 ) G is gravitational constant = 6.673e -11 Nm 2 /kg 2

Find the Fg exerted on the moon (m=7.36e22 kg) by Earth (m=5.98e24 kg) when the distance between them is 3.84e8 m Find the distance between a 0.30 kg ball and a 0.40 kg ball if the magnitude of the Fg is 8.92e-11 N Newton’s Law of Gravitation

Newton’s Law of Universal Gravitation The value of G tells us that gravity is a very weak force. It is the weakest of the presently known four fundamental forces. We sense gravitation only when masses like that of Earth are involved.

Cavendish’s first measure of G was called the “Weighing the Earth” experiment. Once the value of G was known, the mass of Earth was easily calculated. The force that Earth exerts on a mass of 1 kilogram at its surface is 10 newtons. The distance between the 1-kilogram mass and the center of mass of Earth is Earth’s radius, 6.4 × 10 6 meters. from which the mass of Earth m 1 = 6 × kilograms.

When G was first measured in the 1700s, newspapers everywhere announced the discovery as one that measured the mass of Earth

Gravitational Field We can regard the moon as in contact with the gravitational field of Earth. A gravitational field occupies the space surrounding a massive body. A gravitational field is an example of a force field, for any mass in the field space experiences a force. Gravitational field strength equals free-fall acceleration

Field lines can also represent the pattern of Earth’s gravitational field. The field lines are closer together where the gravitational field is stronger. Any mass in the vicinity of Earth will be accelerated in the direction of the field lines at that location. Earth’s gravitational field follows the inverse- square law. Earth’s gravitational field is strongest near Earth’s surface and weaker at greater distances from Earth

Field lines represent the gravitational field about Earth

Weight changes with location Gravitational mass equals inertial mass On the surface of any planet, the value of g, as well as your weight, will depend on the planet’s mass and radius Your weight is less at the top of a mountain because you are farther from the center of Earth. Applications of Gravity

Newton’s law of gravitation accounts for ocean tides. High and low tides are partly due to the gravitational force exerted on Earth by its moon. The tides result from the difference between the gravitational force at Earth’s surface and at Earth’s center. The moon’s attraction is stronger on Earth’s oceans closer to the moon, and weaker on the oceans farther from the moon. This is simply because the gravitational force is weaker with increased distance.

The two tidal bulges remain relatively fixed with respect to the moon while Earth spins daily beneath them.

Earth’s tilt causes the two daily high tides to be unequal.

Kepler’s Laws of Planetary Motion Newton’s law of gravitation was preceded by Kepler’s laws of planetary motion. Kepler’s laws of planetary motion are three important discoveries about planetary motion made by the German astronomer Johannes Kepler.

Kepler started as an assistant to Danish astronomer Tycho Brahe, who headed the world’s first great observatory in Denmark, prior to the telescope. Using instruments called quadrants, Brahe measured the positions of planets so accurately that his measurements are still valid today. After Brahe’s death, Kepler devoted many years of his life to the analysis of Brahe’s measurements.

Kepler’s laws were developed a generation before Newton’s law of universal gravitation. Newton demonstrated that Kepler’s laws are consistent with the law of universal gravitation. The fact that Kepler’s laws closely matched observations gave additional support for Newton’s theory of gravitation.

Kepler’s laws describe the motion of the planets. First Law: Each planet travels in an elliptical orbit around the sun, and the sun is at one of the focal points. Second Law: An imaginary line drawn from the sun to any planet sweeps out equal areas in equal time intervals. Third Law: The square of a planet’s orbital period (T 2 ) is proportional to the cube of the average distance (r 3 ) between the planet and the sun.

Kepler’s 1 st Law Kepler’s expectation that the planets would move in perfect circles around the sun was shattered after years of effort. He found the paths to be ellipses.

Kepler’s 2 nd Law According to Kepler’s second law, if the time a planet takes to travel the arc on the left (∆t 1 ) is equal to the time the planet takes to cover the arc on the right (∆t 2 ), then the area A 1 is equal to the area A 2. Thus, the planet travels faster when it is closer to the sun and slower when it is farther away

After ten years of searching for a connection between the time it takes a planet to orbit the sun and its distance from the sun, Kepler discovered a third law. Kepler found that the square of any planet’s period (T) is directly proportional to the cube of its average orbital radius (r). Kepler’s third law states that T 2  r 3. The constant of proportionality is 4  2 /Gm, where m is the mass of the object being orbited.

Board Work Magellan was the first planetary spacecraft to be launched from a space shuttle. During the spacecraft’s fifth orbit around Venus, Magellan traveled at a mean altitude of 361km. If the orbit had been circular, what would Magellan’s period and speed have been?

Kepler was the first to coin the word satellite. He had no clear idea why the planets moved as he discovered. He lacked a conceptual model. Kepler was familiar with Galileo’s concepts of inertia and accelerated motion, but he failed to apply them to his own work. Like Aristotle, he thought that the force on a moving body would be in the same direction as the body’s motion. Kepler never appreciated the concept of inertia. Galileo, on the other hand, never appreciated Kepler’s work and held to his conviction that the planets move in circles.

Rotational Motion Rotational and translational motion can be analyzed separately. For example, when a bowling ball strikes the pins, the pins may spin in the air as they fly backward. The pins have both rotational and translational motion. Measure the ability of a force to rotate an object.

Torque Torque is a quantity that measures the ability of a force to rotate an object around some axis. How easily an object rotates on both how much force is applied and on where the force is applied. The perpendicular distance from the axis of rotation to a line drawn along the direction of the force is equal to d sin  and is called the lever arm.  = Fd sin  torque = force  lever arm

The applied force may act at an angle. However, the direction of the lever arm (d sin  ) is always perpendicular to the direction of the applied force, as shown here.

In each example, the cat is pushing on the door at the same distance from the axis. To produce the same torque, the cat must apply greater force for smaller angles.

Sign of Torque Torque is a vector quantity. In this textbook, we will assign each torque a positive or negative sign, depending on the direction the force tends to rotate an object. We will use the convention that the sign of the torque is positive if the rotation is counterclockwise and negative if the rotation is clockwise

Board Work A basketball is being pushed by two players during tip-off. One player exerts an upward force of 15 N at a perpendicular distance of 14 cm from the axis of rotation. The second player applies a downward force of 11 N at a distance of 7.0 cm from the axis of rotation. Find the net torque acting on the ball about its center of mass.

Simple Machines A machine is any device that transmits or modifies force, usually by changing the force applied to an object. All machines are combinations or modifications of six fundamental types of machines, called simple machines. These six simple machines are the lever, pulley, inclined plane, wheel and axle, wedge, and screw

Because the purpose of a simple machine is to change the direction or magnitude of an input force, a useful way of characterizing a simple machine is to compare the output and input force. This ratio is called mechanical advantage. If friction is disregarded, mechanical advantage can also be expressed in terms of input and output distance.

In the first example, a force (F1) of 360 N moves the trunk through a distance (d1) of 1.0 m. This requires 360 Nm of work. In the second example, a lesser force (F2) of only 120 N would be needed (ignoring friction), but the trunk must be pushed a greater distance (d2) of 3.0 m. This also requires 360 Nm of work.

The simple machines we have considered so far are ideal, frictionless machines. Real machines, however, are not frictionless. Some of the input energy is dissipated as sound or heat. The efficiency of a machine is the ratio of useful work output to work input. The efficiency of an ideal (frictionless) machine is 1, or 100 percent. The efficiency of real machines is always less than 1.