6. (40 pt) There is a pendulum in Ross Hall. Consider that the length of the wir

Question

6. (40 pt) There is a pendulum in Ross Hall. Consider that the length of the wire is L and the mass of the spherical object is m. Let's compute the length of L. The angle of the location of the spherical mass and the location at the bottom is . Consider er and e, to be the unit vector pointing toward the spherical mass and the tangential vector, respectively (a) (10 pt) Write the equation of motion along en assuming that is small enough to consider sin 0. (b) (10 pt) Describe the velocity in the tangential direction at , assuming-Bo at 0 as the initial condition (c) (10 pt) Compute the period of this pendulum.

Explanation / Answer

6. given mass of spherical object = m

length of pendulum = L

angle with the vertical at given tiume = theta

er = radial unit vector

et = tangential unit vector

a. equation of motion aloing et

mgsin(theta) = ma ( where a is tangential acceleration)

for small theta

mg*theta = ma

g*theta = a ( tangential acceleration)

b. now let tangential velocity be v

then a = dv/dt = g*theta

but v = L*d(theta)/dt

hence

dv*d(theta)/dt*d(theta) = g*theta

(v/L)dv/(d(theta)) = g*theta

vdv = Lg*theta*d(theta)

integrating

v^2 = Lg*theta^2 + c

at theta = 0

d(theta)/dt = thetao' = v/L

hence

(L*thetao')^2 = c

hence

v^2 = Lg*theta^2 + L^2*thetao'^2

c. time period of pendulum

T = 2*pi*sqroot(L/g)

d. for time period of 2 second

2 = 2*pi*sqroot(L/g)

L = 0.9939 m

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