At a certain harbor, the tides cause the ocean surface to rise and fall a distan

Question

At a certain harbor, the tides cause the ocean surface to rise and fall a distance d (from highest level to lowest level) in simple harmonic motion, with a period of 12.4 h. How long does it take for the water to fall a distance 0.250d from its highest level?
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In the figure, a block weighing 15.5 N, which can slide without friction on an incline at angle  = 30.0°, is connected to the top of the incline by a massless spring of unstretched length 0.410 m and spring constant 125 N/m. The block is initially at its equilibrium position. (a)How far from the top of the incline is the block’s equilibrium point? (b)If the block is pulled slightly down the incline and released, what is the period of the resulting oscillations?

Explanation / Answer

Here,

Period , T = 12.4 hour

as for the harmonic motion

h = d * cos(2pi * t/T)

for h = 0.250 * d

0.250 * d = d * cos(2pi * t/T)

cos(2pi * t/T) = 0.250

solving for t

t = 2.60 hour

the time taken to reach this level is 2.60 hours

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theta = 30 degree

W1 = 15.5 N

k = 125 N/m

a) for the eqilibrium distance x

k * x = m * g * sin(theta)

125 * x = 15.5 * sin(30)

x = 0.062 m

the equilibrium point iss 0.062 m from the top.

b)

Time period = 1/(2pi) * sqrt(w/(k * g))

Time period = 1/(2pi) * sqrt(15.5/(125 * 9.8))

Time period = 0.0179 s

the Time period of the oscillations is 0.0179 s

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