Chapter 7
Circular Motion and Gravitation
7.1 Circular Motion Centripetal Acceleration Any object that rotates on a single axis is in circular motion. The axis of rotation is a line perpendicular to the side of the circular path passing through the center of the axis or center. Objects that are in motion have a speed described as the tangential speed (vt) This speed can be measured from the tangent to the circular motion. If this speed is constant then the motion is uniform circular motion. This speed does depend on the distance the object is from the center of the circle. If we think of two horses side by side on a carousel; both are completing a full circle at the same time. The outside horse is covering more distance, so it has a higher tangential speed. Even though an object, like a seat on a farris wheel, may have a uniform motion is still has acceleration due to change in direction. The direction goes back to the equation, which is Δv/Δt, velocity depends on direction. This acceleration is called centripetal acceleration, the acceleration directed toward the center of a circular path.
vt 2 tan gential speed ac centripetal acceleration r radius of path The reason that the object travels in a uniform circular motion is because it is influenced by centripetal acceleration, which means center seeking. The change in velocity pulls the object toward the center. Sample Problem A A test car moves at a constant speed around a circular track. If the car is 48.2m from the track’s center and has a centripetal acceleration of 8.05 m/s2, what is the car’s tangential speed? r= 48.2 m ac= 8.05m/s2
vt 2 ac vt ac r r
8.05m / s 48.2 2
vt 388.01m2 / s 2 19.7 m / s Acceleration due to a change in speed is tangential acceleration. An object, like a car, has tangential acceleration only if it is changing speed as it moves in a circular path. If the car is not changing speed it only has a component of centripetal acceleration. Centripetal Force This has to consider mass and the along with tangential speed and radius. The vertical component that pulls you toward the center of the circular path equal to the gravitational force. This horizontal component is equal to a net force and thus gives us the centripetal force. Newton’s 2 nd law can also be applied as Fc= mac.
mvt 2 Centripetal force: Fc r Sample problem B: A pilot is flying a small plane at 56.6 m/s in a circular path with a radius of 188.5 m. The centripetal force needed to maintain the plane’s circular motion is 1.89x104 N. What is the plane’s mass?
mvt 2 Fc r
4 Fc r 1.89 x10 N 188.5m 3562650 Nm 1112.1kg m 2 2 vt 3203.56 m2 / s 2 56.6 m / s
7.2 Newton’s Law of Universal Gravitation Isaac Newton was the scientist that recognized that the same force that kept planets in orbit around the sum was also responsible for the attraction of objects to the center of the Earth. He called this force of attraction between objects gravitational force. To make is assumption Newton considered the path of a projectile. He thought of the difference in trajectories based on the initial speed of a
cannon ball shot from cannon. He noticed that if the cannon ball was shot with the correct speed it would have a trajectory the same curvature as the Earth. This told him that it would actually orbit the Earth for a short period of time- the gravitational force between the cannon ball and the Earth was actually a centripetal force that made the ball orbit. Newton also quantitatively described gravitational force by relating it to radius and mass.
Fg G
m1m2 , G beign a constant that was later experimentally found to r2
be 6.673x10-11Nm2/kg2. We will use the distance between the two masses centers for our radius. This force also applies to Newton’s 3rd law; there are equal and opposite forces present. In this case the moon exerts a force, gravitational and centripetal, on the Earth. Gravitational force exists between all objects no matter of size. Because we now understand the relationship between mass and acceleration we can relate that to this concept. If the object is of large mass then the object will not “appear” to be accelerating, but a smaller massed object will have an apparent acceleration. On the example earlier of the cannon ball and the Earth: they have gravitational force that attracts them to each other, the object that appears to be accelerating would be the cannon ball because it has a much smaller mass compared to the Earth. Earth will always accelerate toward objects because it has a gravitational force, we may not be able to recognize it. Sample Problem C: Find the distance between a 0.300 kg billiard ball and a 0.400 kg billiard ball if the magnitude of the gravitational force between them is 8.92x10-11N.
Fg G
.300kg .400kg 0.3 m m1m2 m1m2 11 r G r 6.673 x 10 Nm r2 Fg 8.92 x1011 N
Applying the Law of Gravitation Tides are related to the law of gravitation. The tides result from the difference in gravitational force between Earth’s center and surface.
Gravity is also a field force that can be explained by g= Fg/m. This value of g at any given point is equal to the acceleration due to gravity. g= 9.8 m/s2 We know from previous chapters that weight is a factor of gravity. We will now have a new equation that will explain why weight changes. This equation will shoe that g is dependent on mass and distance. With this new equation the free-fall of an object will now take into consideration the mass of the Earth and the radius. The mass of the object does not matter because it will cancel out of the formula. This makes the inertial mass and the gravitational mass equal. The greater the mass of an object the greater gravitational acceleration it has. It also affects the way the object will accelerate. This also tells us that all masses fall with the same acceleration; 9.8m/s2.
g
Fg r2
Gm1m2 mE G r2 r2
7.3 Motion in Space Kepler’s Laws: Until the middle of the 16th century it was thought that the planets moved in circular paths around the Earth. But, this theory did not fit consistently with observed planetary motion. Nicholas Copernicus proposed in 1543 that the sun was the center of the universe and the Earth and several other planets orbited it. The astronomer Johannes Kepler worked with the Copernican theory to develop some clear facts. From this research we get 3 laws. 1. Each planet travels in an elliptical orbit around the sun, and the sun is one of the focal points. 2. An imaginary line drawn from the sun to any plant sweeps out equal areas in equal time. 3. The square of a planet’s orbit period (T2) is proportional to the cube fo the average distance (r3) between the planet and the sun, or T 2 r 3 . The first law stated that planets travel in elliptical orbits not circles. The second law states that if the time a planet takes to travel the arc on the left is equal to the time the planet takes to cover the arc on the right, then Area 1 is equal to Area 2. For the third law, the orbital period (T) is the time a planet
takes to finish one full revolution, and the distance (r) is the mean distance between the planet and the sun. 7.4 Torque and Simple Machines Rotational Motion: The object that spins will have a rotational and linear motion. These two types of motion can be looked at separately. We will isolate rotational motion and measure the ability of force that it takes to rotate the object. If an object rotates on a hinge, it has an axis of rotation. The ability of a force to rotate an object around some axis is measured by a quantity is torque. When an object is being opened the ease at which it opens depends on how much force is applied and where the force is applied. More torque is produced the farther the force is from the axis of rotation. Think about a door. Where is the door knob? It is on the opposite side of the door from the hinges. The perpendicular distance from the axis of rotation to line drawn along the direction of the force is a lever arm. The lever arm also depends on the angle of the force exerted on the object. If the angle is less than 90 degrees the door will still rotate it will just be more difficult to rotate.
Fd sin This is the force times the lever arm. SI unit of Nm.
The distance of the lever arm is calculated by the distance from the axis of rotation to the point where force is applied. The dsinθ is the line perpendicular distance from the axis of distance from the axis of rotation to a line drawn along the direction of force. The sign of torque: Torque is a vector quantity. Depending on the direction of force it tends to rotate about the object torque will be assigned a sign of positive or negative. Sample Problem E: A basketball is being pushed by two players during tip-off. One player exerts an upward force of 15N at a perpendicular distance of 14 cm from the axis of rotation. The second player applies a downward force of 11N at a perpendicular distance of 7.0 cm for the axis of rotation. Find the net torque acting on the ball about its center of mass.
F1= 15N, d1= 14 cm, F2= 11N, d2= 7 cm
net Fd ,1 2 F1d1 F2 d2
net 15 N 0.14m 11N 0.07m 2.1 Nm 0.77 Nm 2.9 Nm Types of Simple Machines: any device that makes tasks easier is a machine. All machines are combonations of 6 fundamental types of machines. They are: lever, pulley, inclined plane, wheel and axle, wedge, and screw. These machines use the mechanical advantage to measure how effect a machine will be. The MA measures the output force and the input force. Output and input can be applied to a hammer removing a nail from the wall. There is an input force applied to the end of the hammer. The output force is the force that is applied to the nail to remove it from the wall. The input force is smaller than the output force. The torque equation can also be applied to the MA equation.
in out , Fin din Fout dout , MA
Fout dout Fin din
This equation can also explain that the longer the input lever arm is compared to the output lever arm, the greater the amount of MA. This indicates the factor by which input is applied. You don’t have to apply as much input as it takes to do something. The small input will equal a large output. Efficiency can tell us how well machines work. Efficiency is the ratio of useful work output to work input. If the machine is frictionless, like everything else we have talked about, the mechanical energy is conserved. This means that input work (work done on machine) is equal to the output work (work done by machine). MA in thin case would be 1 or 100%. This the best efficiency that you can have. But most machines will encounter some sort of friction and thus decreasing their efficiency. Real machines will have efficiency of less than 1 or 100%.
Wout eff To calculate: Win
