There are a total of three questions. Verify that the formula u(t) = A cos(omega

Question

There are a total of three questions.

Verify that the formula u(t) = A cos(omega 0 t + ) is a solution to the differential equation for the mass on a spring, by plugging this expression for u(t) directly into the differential equation (91). A mass suspended by a spring stretches an additional 5 cm when an additional 10 gram mass is attached to it. What is the value of the spring constant k, in SI units? Calculate the angular frequency, the frequency, and the period of a simple pendulum consisting of a bob of mass of 0.1 kg suspended by a massless string of length 0.5 m. d2u / dt2 + = 0

Explanation / Answer

1.)

du/dt = -Asin(t+)
d2u/dt2 = -A2cos(t + )

put "u" and "d2u/dt2" in the left side of given equation:

-A2cos(t + ) + 2Acos(t + )
= 0
Hence verified! :)

2.)
let mass be m in kg. so,
Initial stretch = mg/k

and according to given condition,
final stretch = initial stretch + .005m

(m + .01)g/k = mg/k + .005

.01g/k = .005
k = .01g/.005 = 2g
That will be in SI units. :)

3.)
Time period, T of such system is 2(l/g), so
T = 2(.5/g)

angular frequency, = 2/T = (g/l) = 2g

frequency = 1/T = (2g)/2

Rate me if you understood properly. :)

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