How does the period of a simple pendulum vary theoretically with the length, the mass of the bob?
816299
the time period = 2 * pi * sqrt (l/g)\n\nthus period is proportional to square root of length.\n\nTheoretically it is not affected by the mass of the bob.
1147420
If you consider a pendulum to be a simple machine, and a spring to be a simple machine.  You could create a 'complex' machine by taking two pendulums and attaching a spring in between the pendulums.  The collective behavior of these 3 simple machines is now significantly more complicated than had they not been coupled together.
795446
f = (1/2pi)sqrt(k/m); where f = frequency of oscillation, m = mass, and k = spring coefficient\n\nThus f' = f [sqrt(k/2m)/sqrt(k/m)] = f sqrt(m/2m) = f sqrt(1/2); where f' is the frequency of the spring with 2m mass.\n\nThe physics lesson is that spring frequencies vary as the square root of the masses. [See source.]
951868
Equate the gravitational force to the centripetal force\n\nFg = mv^2/r\n\n(GMm)/r^2 = mv^2/r, solve for v\n\nv = sqrt(GM/r)\n\nThe period is (2Pi)r/v = (2Pi)rsqrt(r/GM)\n\nwhere r = radius of earth + distance to orbit\nG = gravitational constant\nM = mass of Earth
574466
A heavy pendulum with a ballpoint
1117824
Pendulums work like this...\n\nhttp://en.wikipedia.org/wiki/Pendulum\n\nNothing magical.  Just simple physics.
827544
1) use conservation of energy:\n\n1/2*m*vb^2=1/2*m*va^2+m*g*h\n\nm=2\nva=4\ng=9.8\nh=6\nv=sqrt(16+2*9.8*6)\n=11.56 m/s\n\n2.\nSame concept\nm*g*(bounce)=1/2*m*va^2+m*g*(drop)\nmass divides out\nbounce=va^2/(2*g)+drop\nva=6\ng=9.8\ndrop=4\nbounce=36/(2*9.8) + 4\n=5.84 m\n\n3)  since the mass isn't moving it's offset the 300N force\n\n4)\nAgain, use conservation of energy\nNote that the change in potential energy will subtract from the kinetic energy of the start \n\n1/2*m*vc^2=1/2*m*va^2-m*g*delta-h\nthe mass divides out\nva=8\ndelta-h=2\ng=9.8\nvc=sqrt(64-2*9.8*2)\n=4.98\n\nj
506358
an energy oscillation
93773
well force is measured in kg*m/s^2 so, force will cause a function of time and distance to increase exponentially over the interval of the force applied.\n\nFor example, an object dropped from 100 meters will go 5 meters in the first second, 20 meters in the 2nd second, and  25 meters after 3 seconds, etc.
229095
mass doesn't change. Only weight changes because weight is mass times the acceleration due to gravity.\n\nWt=mg
879231
The simple pendulum requires a massless connection between the pivot and the oscillating mass.  Also, the mass must be concentrated at one discrete radius (can have no radial thickness).  Add air and pivot drag to these and you essentially have the difference between theory & reality........
986622
As mentioned before, frequency increases, and period decreases.  The reason why for both parts is because it's traveling a shorter distance back and forth...so, the frequency (how often it passes a particular point, like the midpoint or ends, in a given amount of time) goes up; and the period (how long it takes to goes from one point to another) is shortened and therefore decreases.
575717
mass * G = space^3 / time^2 \n\nOr more fundamentally: \ntime(mass) = time(interval)^3 / time(period)^2 \n\n( mass * G / C^3 = time(interval)^3 / time(period)^2 ) \n\nC is a constant that is used to differentiate \nbetween time periods (Auto-correlations), \nand time intervals (Cross-correlations). \n\nG is a constant that arises because of the limitations \nof two body math, and is used \nand bodies perceived to be varying in media. \n( Media = time and space constructs.)
94009
Weight can be determined by mass and the pull of gravity exerted on an object. But mass is not an indication of weight. Mass is presumed to be constant through varying gravities while weight fluctuates because it is proportional to both of these variables.
763829
I'm not familiar with that particular book, but the differential equations of simple harmonic motion are interesting.  You can make the system as complex as you like.  I've studied SHM in both maths and physics.  I simulated SH oscillators with operational amplifiers; in other words an analogue computer.  First you simulate a frictionless pendulum, then you introduce friction so the oscillations die away, then you introduce forcing terms of various frequencies to see what happens; like pushing a child on a swing.  If you're really clever, you can introduce hysteresis (delay), and extra degrees of freedom (flexible joints in the pendulum), and vary the Q of the system (how sharply you can tune it to the resonant frequency).  It's about the most open-ended project I ever saw.  I did it in second-year physics.  At its simplest, it's just sixth form maths, but you can turn it into an impossibly complex exercise in chaotic systems if you're clever.
820986
1.\nPhases:\nf1 = (t/T1)*2*pi, where T1 = 1 sec\nf2 = (t/T2)*2*pi, T2 = 1 sec\nIn phase: f1 = f2 + 2*pi*n, where n is integer (0, +-1, +-2, ...)\n(t/T1)*2*pi = (t/T2)*2*pi + 2*pi*n\nt(1/T1 + 1/T2) = n\nT2 = 0.98 = 98/100 = 49/50, 1/T2 = 50/49\nt(49/49 - 50/49) = n\n-t/49 = n\nt = 49 seconds, n = -1\nNumber of oscillations if second, shorter pendulum (shorter pendulum has shorter period): N = t/T2 = 49/(49/50) = 50\n\n2.\nCoordinate: y(t) = 3*sin(20*pi*t)\nSpeed: v(t) = y'(t) = 20*pi*3*cos(20*pi*t)\nMean position: y(t) = 0. Take t0 = 0.\nMaximum displacement: y(t1) = amplitude = 3, speed = v(t1) = 0\nv(t1) = 0\ncos(20*pi*t1) = 0\n20*pi*t1 = pi/2\nt1 = 1/(2*20) = 1/40
1005080
Oscillation, in general, is a periodic fluctuation between two things; in the broadest sense, oscillation can occur in anything from a person's decision-making process to tides and the pendulum of a clock. Oscillation in a device called an oscillator is usually a back and forth motion over a central neutral point, created by changes in energy. In a pendulum-driven clock, for example, the oscillation is the back and forth movement of the pendulum. Oscillators may be mechanical or electronic, but all work on the same principles. Other devices based on the principles of oscillation include the oscillograph and the oscilloscope. \nLike other oscillators, a clock pendulum's oscillation is maintained by changes in energy. In this case, potential energy, present when the pendulum is at the top of its swing, is converted to kinetic energy as the pendulum falls and is driven upwards on the other side. When the kinetic energy has been spent, at the top of the swing, the pendulum's energy is potential once more. With no kinetic energy to drive it higher, the pendulum falls. A pendulum clock keeps time according to the frequency of the pendulum's swing (the number of times it swings per second). Friction would eventually cause the movement to stop, but mechanical pendulum clocks use a spring to help the device overcome friction's drag. Most modern timepieces use quartz or electronic oscillators. The most accurate timepiece in the world, the atomic clock, measures time according to the oscillation within atoms.