#$&* course PHY 121 8/2 11 034. Simple Harmonic Motion*********************************************
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Given Solution: Since `omega = `sqrt( k / m), we have `omega = `sqrt( (36 N/m) / (4 kg) ) = `sqrt( 9 (N/m) / kg ) = `sqrt( 9 [ (kg m/s^2) / m ] / kg ) = `sqrt(9 s^-2) = 3 rad/s. Always remember that this quantity stands for the angular velocity of the point on the reference circle. [ There is a good reason why we get the radian unit here, but to understand that reason requires a very good understanding of calculus so we're not going to discuss it at this point.] A cycle of pendulum motion corresponds to a complete trip around the circumference of the circle, an angular displacement of ` pi radians. So if the reference point is moving around the circle at 3 rad/s, to complete a cycle of 2 `pi rad requires time T = 2 `pi rad / (3 rad/s) = 2 `pi / 3 sec, or approximately 2.09 sec. This time is called the Period of Motion of the pendulum, and is customarily designated T. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q002. Recall that a pendulum with mass m and length L experiences a restoring force F = - m g / L * x, so that we have F = - k x with k = m g / L. What is the period of motion of a pendulum of length 3 meters and mass 10 kg? What would be the period of a pendulum of length 3 meters and mass 4 kg? Does your result suggest a conjecture? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I see enough information to find 'k' as follows: k = 10 kg * 9.8 m/s^2 / 3 m = 98 kg m/s^2 / 3 m k = 32.67 kg/s^2 You can use k to find the velocity: `omega = `sqrt k/m `omega = `sqrt 32.67 kg/s^2/10 kg `omega = `sqrt 3.267 radians^2/s^2 `omega = 1.807 radians/s And, now I'm stuck. I'm thinking that the radius might be proportional or equal to the length of the pendulum, but I don't know, so that I can find a period of time. That much for the second one: K = 4 kg (9.8 m/s^2) / 3 m K = 39.2 kg m/s^2 / 3 m K = 13.067 kg/s^2 Finding velocity: `omega = `sqrt k/m `omega = `sqrt 13.067 kg/s^2 / 4 kg `omega = `sqrt 3.26675 radians^2/s^2 `omega = 1.807 radians/s Conjecture: The velocity depends on the length of the pendulum and not the mass put on it. If you put it all together, you will be taking the square root of mass times gravity divided by length, all divided by mass. The two mass numbers will cancel each other out, so the velocity is really the square root of gravity divided by length of pendulum. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: For a pendulum 3 meters long with mass 10 kg, we have k = m g / L = 10 kg * 9.8 m/s^2 / (3 meters) = 32.7 ( kg m/s^2 ) / m = 32.7 N / m. The angular velocity of the reference point for this pendulum is thus `omega = `sqrt( k / m ) = `sqrt ( 32.7 N/m / (10 kg) ) = `sqrt( 3.27 s^-2) = 1.81 rad/s. For a pendulum 3 meters long with mass 4 kg we have k = m g / L = 4 kg * 9.8 m/s^2 / (3 meters) = 13.1 N / m, so `omega = `sqrt( 13.1 N/m / (4 kg) ) = `sqrt( 3.28 s^-2) = 1.81 rad/s. These angular frequencies appear to be the same; the only difference can be attributed to roundoff errors. This common angular frequency implies a period T = 2 `pi / `omega = 2 `pi / ( 1.81 rad/s ) = 3.4 sec, approx.. Noting that both pendulums have length 3 meters we therefore conjecture that any pendulum of length 3 meters will have an angular frequency of 1.81 radians/second and period approximately 3.4 sec. We might even conjecture that the period of a pendulum depends only on its length and not on its mass. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I came closer than I thought. I have written down 2 `pi divided by `omega = elapsed time in my notes. ------------------------------------------------ Self-critique rating: 2 ********************************************* Question: `q003. What is a symbolic expression for the period of a pendulum of length L and mass m? Hint: Follow the same reasoning steps as in the preceding example, but instead of numbers use symbols at each step. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find k, you would get m*g/L Next you would solve for `omega: `sqrt k/m Finally, you would solve for elapsed time 2 `pi / `omega If you put it all together into one problem it gets a bit mess, but it would look like this: 2 `pi / `sqrt [(m*g/L)/m)] The masses would actually cancel each other out, and you would have: 2 `pi / `sqrt g/L confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The reasoning process went like this: We found the restoring force constant k from the length and the mass, obtaining k = m g / L. Then we found the angular frequency `omega = `sqrt( k / m ) using the value we obtained for k. Our result here is therefore `omega = `sqrt( k / m ) = `sqrt( [ m g / L ] / m ) = `sqrt( g / L ). We note that the mass divides out of the expression so that the angular frequency is independent of the mass. The period is T = 2 `pi / `omega = 2 `pi / (`sqrt ( g / L ) ) = 2 `pi `sqrt( L / g ). [ If you don't see what's going on in the last step, here are the details: 2 `pi / `sqrt( g / L ) = 2 `pi / [ `sqrt(g) / `sqrt(L) ] = 2 `pi * `sqrt(L) / `sqrt(g) = 2 `pi `sqrt( L / g ) ]. Our expression for the period is also independent of the mass. This would confirm our conjecture that the period of a pendulum depends only on the length of the pendulum and is independent of its mass. NEARLY CORRECT STUDENT SOLUTION: k = mg/L omega = sqrt(k/m) period of pendulum = 2pi rad / omega INSTRUCTOR RESPONSE: Your expressions are correct, but your answer isn't in terms of L and m. Using your expressions, it's easy to get a result in terms of L and m. k = m g / L so omega = sqrt(k / m) = sqrt( (m g / L) / m) = sqrt( g / L). So period of pendulum = 2pi rad / omega = 2 pi / (sqrt(g / L ). This answer would be acceptable for a Physics 121 student, but the denominator involves a fraction, so it isn't quite in standard form. 2 pi / (sqrt(g / L ) = 2 pi * sqrt( L / g), which is standard form. STUDENT QUESTION: I knew you had to obtain omega somehow. I got a little confused on how omega was equal to sqrt(g/L). Could you explain that a little more INSTRUCTOR RESPONSE: The net force on a pendulum displaced x units from equilibrium, where | x | < < L (magnitude of displacement is much less than pendulum length), is F_net = - (m g / L) * x. This was found in previous assignment, using the fact that the vertical component of pendulum tension is very close to m g, and is in the same proportion to the horizontal component as the displacement to the length of the pendulum. Thus F_net = - k x, where k = m g / L. Since omega = sqrt(k / m), we have omega = sqrt( (m g / L) / m) = sqrt(g / L). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: 2 ********************************************* Question: `q004. The frequency of a pendulum is the number of cycles completed per unit of time. The usual unit of time is the second, so the frequency would be the number of cycles per second. What is the frequency of a pendulum of length 20 cm? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: `omega = `sqrt k/m `omega = `sqrt m*g/L / m `omega = `sqrt g/L `omega = `sqrt 9.8 m/s^2 / .2 m `omega = `sqrt 7 s^2 `omega = 7 radians/s `dt = 2 `pi rad/`omega `dt = 2 `pi rad/ 7 rad/s `dt = .898 s I know there needs to be something more, but I can't figure out what it is.... confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We know that the period of a pendulum is T = 2 `pi `sqrt( L / g ). Using L = 20 cm we must use g = 980 cm/s^2 in order to have compatible units in our calculation, we obtain T = 2 `pi `sqrt( L / g ) = 2 `pi `sqrt( 20 cm / (980 m/s^2) ) = 2 `pi `sqrt( .02 s^-2) = 2 `pi * .14 rad/sec = .88 sec (approx). The period represents the number of seconds required for the pendulum to complete a cycle. To obtain the frequency, which is the number of cycles per second, we take the reciprocal of the period: f = 1 / T = 1 / (.88 sec / cycle) = 1.14 cycles / sec. This pendulum will go through 1.14 complete cycles in a second. STUDENT COMMENT: i didn't know where to start here, i see how you got the answer for T, but had no idea to get frequency by dividing 1 by T INSTRUCTOR RESPONSE: Be sure to make a note of this. It's not difficult to remember, but if you think in terms of the meanings of the quantities it's easier than trying to keep a bunch of similar-looking formulas straight. Recall the definitions of frequency and period from the very first experiment of the course, the Introductory Pendulum Experiment. It was not an accident that we started the course with the very last, and perhaps most challenging topic. We have encountered the pendulum at many points in the course, and these last few assignments are designed to complete our understanding of this common system. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I did better than I thought I would. I wrote the reciprocal part in my notes. ------------------------------------------------ Self-critique rating: 2 ********************************************* Question: `q005. Early in the course the period of a pendulum was said to be related to its length by the equation T = .20 `sqrt(L), where T is in seconds when L is in cm. If we rearrange the equation T = 2 `pi `sqrt( L / g ) to the form T = [ 2 `pi / `sqrt(g) ] * `sqrt(L) and express g as 980 cm/s^2, we can simplify the factor in brackets. Do so and explain how your result confirms the equation given earlier in the course. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If you multiply 2 `pi, using 3.14 as `pi, you will get 6.28. Dividing that by the square root of 980 cm/s^2, you end up with .2006. This would give you .2 * `sqrt L. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The factor in brackets is [ 2 `pi / `sqrt(g) ], which becomes 2 `pi / `sqrt(980 cm/s^2) = 2 `pi / ( 31.3 `sqrt(cm) / s ) = .20 s / `sqrt(cm). The equation is therefore T = .20 s / `sqrt(cm) * `sqrt(L). If L is given in cm then `sqrt(L) will be in `sqrt(cm) and the units of the calculation will be seconds. STUDENT COMMENT: this factoring and making equations confuse me, it is hard enough to figure out the number to go into given equations INSTRUCTOR RESPONSE: Such confusion is typical among Principles of Physics students, is not unusual among General College Physics and even occurs among students in University Physics. When the expressions get complicated, be sure you write them out on paper. You can't hope to read or operate on a challenging algebra expression without writing it down in standard form. The algebra involved here approaches but does not exceed the level of a high school Algebra II class and should be