You want to make an old-fashioned pendulum clock, with a period of exactly 1.5 s

Question

You want to make an old-fashioned pendulum clock, with a period of exactly 1.5 second(s). Use the Earth's gravitational acceleration g = 9.8 m/s2.

a) Ignoring the size of the pendulum bob, from what length wire should you hang your pendulum bob?
m

b) If you release the bob from an angle of 3.8°, what is the approximate acceleration of the bob at release? (you can use the approximation, sin(?) = ?)
m/s2

c) Describing a pendulum as a harmonic oscillator requires the use of the small angle approximation. What is the magnitude of the difference between the actual acceleration (without using the small angle approximation) and the approximate acceleration (using the small angle approximation)? The answer is a very small number. As for all other numerical answers, provide your answer with 4 significant digits.
m/s2

d) From this release angle, what is the change in the bob's height from release to the bottom of its arc?
m

e) What is the bob's speed at the bottom of its arc? (use conservation of energy)
m/s

Explanation / Answer

a)       Given that,       The time period of the pendulum, T = 1.5 s       The given angle, = 3.80                The time period of the pendulum is                             T = (2)(L/g)       here, L = length of the pendulum                g = acceleration due to gravity = 9.8 m/s2 _____________________________________________________________________________ _____________________________________________________________________________       Therefore, length of the pendulum is given by,                    L = T2g/42                      = (1.5 s)2(9.8 m/s2 ) / 4()2                        = 0.559 m                      = 0.56 m                      = 0.56 m _____________________________________________________________________________ _____________________________________________________________________________ b)       We have that,       The acceleration of pendulum is                      a = gsin       For small oscillations, sin =       Therefore, a = g       Then the acceleration is                                a = (9.8 m/s2 )(3.80)                                   = (9.8 m/s2)(3.8/180)                                   = 0.649631 m/s2 ______________________________________________________________________ ______________________________________________________________________ c)      Actual acceleration, a = gsin                                          = (9.8 m/s2 )sin3.80                                           = 0.649484 m/s2       The difference between actual acceleration and acceleration is                 a = 0.649631 m/s2 - 0.649484 m/s2                       = 0.000147 m/s2 ________________________________________________________________________ ________________________________________________________________________
d)
      Height of the pendulum from its bottom of release is,                         h = L(1-cos )                           = (0.56 m)(1-cos3.80)                           = 0.00123 m _______________________________________________________________________ _______________________________________________________________________
e)
      From the law of conservation of energy ,       Kinetic energy = potential energy              (1/2)mv2 = mgh                          v = 2gh                             = (2)(9.8 m/s2)(0.00123 m)                             = 0.155 m/s       For small oscillations, sin =       Therefore, a = g       Then the acceleration is       Then the acceleration is                                a = (9.8 m/s2 )(3.80)                                   = (9.8 m/s2)(3.8/180)                                   = 0.649631 m/s2 ______________________________________________________________________ ______________________________________________________________________ c)      Actual acceleration, a = gsin                                          = (9.8 m/s2 )sin3.80                                           = 0.649484 m/s2       The difference between actual acceleration and acceleration is                 a = 0.649631 m/s2 - 0.649484 m/s2                       = 0.000147 m/s2 ________________________________________________________________________ ________________________________________________________________________
d)
      Height of the pendulum from its bottom of release is,                         h = L(1-cos )                           = (0.56 m)(1-cos3.80)                           = 0.00123 m _______________________________________________________________________ _______________________________________________________________________
e)
      From the law of conservation of energy ,       Kinetic energy = potential energy              (1/2)mv2 = mgh                          v = 2gh                             = (2)(9.8 m/s2)(0.00123 m)                             = 0.155 m/s                                          = (9.8 m/s2 )sin3.80                                           = 0.649484 m/s2                                           = 0.649484 m/s2       The difference between actual acceleration and acceleration is                 a = 0.649631 m/s2 - 0.649484 m/s2                       = 0.000147 m/s2                       = 0.000147 m/s2 ________________________________________________________________________ ________________________________________________________________________
d)
      Height of the pendulum from its bottom of release is,                         h = L(1-cos )                           = (0.56 m)(1-cos3.80)                           = 0.00123 m                           = 0.00123 m _______________________________________________________________________ _______________________________________________________________________
e)
      From the law of conservation of energy ,       Kinetic energy = potential energy              (1/2)mv2 = mgh                          v = 2gh                             = (2)(9.8 m/s2)(0.00123 m)                             = 0.155 m/s                             = 0.155 m/s

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