081105
Activity A:
Rotate the strap on the die and time it from the instant of release until it comes to rest, while also determining the number of degrees through which it rotates. Do this for several trials (as many as you can conduct in 10 minutes).
Find the average rate of change of angular position (in degrees) with respect to clock time (in seconds) for each trial.
Terminology and symbols:
The Greek letter theta (symbol q) denotes angular position. Theta can be measured in degrees, but as we will see it is often easier to measure theta in radians (which will be defined below).
The Greek letter omega (symbol w) denotes angular velocity, which is rate of change of angular position with respect to clock time. You found the average angular velocity above.
The Greek letter alpha (symbol a) denotes angular acceleration, which is rate of change of angular velocity with respect to clock time.
Continued activity A:
For the first of your trials, assume that the graph of angular velocity vs. clock time (i.e., omega vs. t) is linear.
Sketch a plausible graph of omega vs. t for the interval you timed. Label the axes but don't yet put a scale on either axis.
Now label all the information you currently have related to the motion on the interval.
Using your graph, find the information you need in order to determine the average rate of change of velocity with respect to clock time for the interval. (Reason out what is being asked, starting with the definition of average rate of change of A with respect to B).
Find the average rate of change of velocity with respect to clock time for each of the intervals you originally timed.
Activity B:
Place a paper clip on each end of the strap.
If you spin the strap too fast the paper clip slides off the end.
If you try to start the strap too fast the paper clip slides off the side. You want to avoid this, so start the rotation gradually.
Do several trials, in about half of which the paper clips stay put as the strap coasts to rest, in the other half of which the paper clips slide off the ends. Take data sufficient to find the initial velocity and acceleration of the coasting strap.
Determine the maximum angular velocity at which the paper clips stay put.
Determine the minimum angular velocity at which the paper clips slide off the ends.
Find the velocity of the paper clips at each angular velocity, in cm / sec.
More information:
The centripetal acceleration of an object moving at velocity v on a circle of radius r is
a_Cent = v^2 / r.
Find the centripetal acceleration of the 'fastest' paper clip that stays put.
Find the centripetal acceleration that would be required to hold in its circular path the 'slowest' paper clip that slides off the end.
Activity C:
Sketch a unit circle--a circle of radius 1 centered at the origin of an x-y coordinate system.
Sketch the vector r directed at 60 degrees, whose initial point is the origin and whose terminal point is on the circle. What are the coordinates of the terminal point of this vector?
Sketch the line tangent to the circle at the terminal point of vector r.
What angle does this tangent line make with the direction of the positive x axis?
What would be the x and y components of a vector in the direction of the tangent line, if the magnitude of the vector is 4?
A vector of magnitude .6 is directed from the terminal point of r back toward the origin. What angle does this vector make with the positive x axis, and what are its x and y components?
Now consider again the motion of the paper clip on the rotating strap. If the strap is rotating at the maximum possible speed that will still allow the paperclip to stay on the strap:
How fast is the clip moving?
Sketch the circle representing the path of the paper clip, with the center of the circle represented by the origin of an x-y coordinate system. Find the following:
The x and y coordinates of the vector r from the center of rotation to the circle, at the instant the paper clip reaches the 60-degree position.
The x and y coordinates of the vector v that represents the velocity of the clip at this instant (hint: how fast is the clip moving, and what angle does its path at this instant make with the positive x axis?).
The x and y coordinates of the vector a that represents the acceleration of the clip at this instant (hint: what is the direction of the acceleration and what is its magnitude?).
Activity D:
The gravitational force exerted by the Earth on an object of mass m is F_grav = G * m * M / r^2, where r is the distance from the center of the Earth to the object, G = 6.67 * 10^-11 N m^2 / kg^2 and M is the mass of the earth, which is about 6 * 10^24 kg. This applies as long as r is greater than the radius of the Earth, which is about 6400 km. Using this information:
How much force does the Earth exert on the 40 kg mass of the table at which you are sitting?
What therefore would be the acceleration of the table if it was raised a short distance off the floor and released?
Is this consistent with the accepted value of g?
What would be the force on a satellite of mass 200 kg, at a distance of 800 km 'above' the surface of the Earth?
What therefore would the acceleration of the satellite toward the center of the Earth, assuming its path to be circular?
In order to have this centripetal acceleration in this path, what velocity would the satellite require?
In general if a satellite is in a circular orbit at distance r from the center of the Earth, what is the expression for its gravitational force, and what is the resulting expression for its gravitational acceleration?
If the satellite is in a circular orbit, then its centripetal acceleration is equal to the gravitational acceleration.
What is the expression for its centripetal acceleration?
What is the expression for its gravitational acceleration?
What equation therefore results from the assumption that the centripetal acceleration is equal to the gravitational acceleration?
Solve this equation for v.
What therefore is the kinetic energy of the satellite?