Physics 124 — Particles and Waves Chapter 5: Dynamics of Uniform Circular Motion: Physics 124 — Particles and Waves Chapter 5: Dynamics of Uniform Circular Motion Frans Pretorius
University of Alberta
Uniform circular motion: Uniform circular motion An object traveling at constant speed on a circular path is said to undergo uniform circular motion. The radius of the path is a constant r
The speed is a constant v (though note that the direction changes!)
The period T is the amount of time needed to complete one revolution. Therefore:
Uniform circular motion: Uniform circular motion Because the direction of the velocity of the object is changing, it is accelerating:
Centripetal Acceleration: Centripetal Acceleration this acceleration is called centripetal acceleration. Its magnitude is and its direction is always toward the center of the circle.
Centripetal Force: Centripetal Force By Newton’s second law, if an object is accelerating, there must be a net external force acting on it. For an object undergoing uniform circular motion, this is called the centripetal force
The direction of the force is the same as the direction of the acceleration, i.e. towards the center of the circle, and the magnitude is:
Note: the centripetal force is sometimes called a “fictitious” force, because in all situations there will be some other force (gravity, friction, tension) that’s responsible for producing the centripetal force
Example A (Ch. 5, problem 19): Example A (Ch. 5, problem 19) A ride at a carnival swings a person in a chair, attached by a cable to a central vertical pole, about the pole at a constant height and with constant speed. The angle the cable makes relative to the pole is 60.0°, the length of the cable is 15.0m, and the combined mass of the chair and person is 179kg . a) What is the tension in the cable? b) Find the speed of the chair. Image courtesy John Wiley & Sons, Inc.
Example A (Ch. 5, problem 19): Example A (Ch. 5, problem 19)
Answers: a) 3510N, b) 14.9m/2
Banked curves: Banked curves Usually when a car turns, friction between the tires and road provided the centripetal force needed to make the turn. However, if the curve in the road is banked at an angle, then with the correct speed for the given angle, gravity and the normal force is all that’s needed: Image courtesy John Wiley & Sons, Inc.
Circular orbits: Circular orbits Consider an object will a small mass m in orbit around a very large mass M. Newton’s law of gravity tells us that both objects exert the same magnitude of force on one another. But if M is much larger than m, then Newton’s second law of motion says that the acceleration of the larger object will be much smaller than that of the small object. Thus, we can consider the large object to be at a fixed location without much loss of accuracy. Applicable to orbit calculations:
a satellite (m) orbiting the earth (M)
the earth (m) orbiting the sun (M)
Circular orbits: Circular orbits Many orbits are possible, given the speed v of the small mass and the distance r to the large mass M
usually classified as elliptic (bound orbits), parabolic (marginally bound) or hyperbolic (unbound). Example … stars in orbit around the suspected black hole at the center of our galaxy http://www.mpe.mpg.de/ir/GC/index.php (all elliptic orbits in this case)
The circular orbit is a special case of an elliptic orbit, and for a given orbital radius there is only a single velocity that generates a circular orbit:
Example B. (ex. 9 from Ch 5): Example B. (ex. 9 from Ch 5) Determine the speed of the Hubble Space Telescope orbiting a distance of 598km above the earth’s surface.
Example B.: Example B.
Answer: 7.56x103m/s
Orbit example: the problem of dark matter in cosmology: Orbit example: the problem of dark matter in cosmology there are at present two fundamental “crisis” in our understanding of large scale structures (i.e. galaxies, clusters of galaxies) in our universe
dark energy (a mysterious “force” that appears to be causing the expansion of the universe to accelerate)
dark matter
galaxies seem to be more massive that what they should be
Dark matter and galaxies: Dark matter and galaxies How heavy should a galaxy be?
from studies of nearby stars, we know how much light a star of a given mass radiates
therefore, can “count” the total number of stars by measuring the total light coming from a galaxy
can estimate how much non-luminous matter (such as dust and gas) there is
add all this up, and astronomers are pretty sure they know how heavy galaxies should be
How do you weigh a galaxy to test that?
orbits! M81 … a spiral galaxy containing billions of stars
Weighing galaxies: Weighing galaxies We’ve just found that the velocity of a circular orbit is given by
In spiral galaxies, most stars follow close to circular orbits, with the mass M in that case being the total mass enclosed within a radius r. So, how fast a star is moving tells us how much mass it is orbiting (which, incidentally, is the same method we use to measure the mass of the earth, moon, sun and planets):
Can measure the velocities of stars via doppler shifts (same way police radar guns work to measure car velocities), and a whole slew of methods can be employed to find the distances involved
Weighing galaxies: Weighing galaxies (from astro-ph/980225) NGC2403 Measured velocities (km/s) of stars as a function of distance from the center of the galaxy (in kiloparsecs) An estimate of what velocities ought to be if only the visible stars were taken into account. i.e, most of the stars are clustered near the center, so the velocities should be decreasing at large r like How then can the actual velocities stay constant with distance? There must be something there that we can’t see but that has mass — dark matter!
Apparent weightlessness and artificial gravity: Apparent weightlessness and artificial gravity Standing on the surface of the earth, the force of gravity is directed downwards. However, what we “feel” as the force of gravity, i.e. our weight, is the normal force of the earth directed upwards against us.
therefore, when you are in free-fall (e.g. skydiving), the force of gravity is still acting on you, and downward, but since there is no normal force, you feel weightless
So to creative artificial gravity in a space station means coming up with a way to reproduce the normal force we are used to
one way to do so is via uniform circular motion
Example C. : Example C. A space station has the shape of a large cylindrical drum, rotating with constant velocity. Astronauts live on the inside surface of the station. If the radius of a circular cross-section is 100m, how many revolutions per minute about the axis of the cylinder must the station make to simulate the force of gravity on earth? (is this scenario exactly the same as gravity? … what happens if the astronaut holds a ball and “drops” it?) Image courtesy John Wiley & Sons, Inc.
Example C.: Example C.
Answer: 3.0 revolutions per minute
Vertical circular motion: Vertical circular motion Here, we’re still looking at motion in a circle, but now gravity is introduced as a force that can add to or reduce the “other” forces that must be exerted on the object to keep it on a circular path (in this example, the other force is the normal force of the track).
the net force in the radial direction is still the centripetal force
In positions (2) and (4), the normal force equals the centripetal force
In position (1) the magnitude of the normal force is equal to the magnitude of the centripetal force plus that of the weight of the object
In position (3) the magnitude of the normal force is equal the magnitude of the centripetal force minus that of the weight of the object Image courtesy John Wiley & Sons, Inc.
Example D.: Example D. A motorcycle driver is riding around on the inside of a vertical track as shown. The radius of the track is 10m. What is the minimum velocity the motorcycle must have not to fall when at the very top of the course?
Example D.: Example D.
Answer: 10m/s (36km/hour)