# Simple harmonic motion: Position velocity and period

A raft is bobbing up and down in a lake. Its vertical position x above the flat surface of the water is given by x=0.60sin(3.0t+4.2), where distances are in metres and time t is in seconds.

1. What is the boat's position at t=0? At t=2.8s?

2. What is the period of the raft's oscillation?

3. What is the boat's maximum speed during this oscillation?

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A raft is bobbing up and down in a lake. Its vertical position x above the flat surface of the water is given by x=0.60sin(3.0t+4.2), where distances are in metres and time t is in seconds.

The raft is simply moving up and down and not rotating hence may be considered as a particle in motion.

As the displacement of the raft is a sine function of time its motion is Simple harmonic motion.

The equation of motion of a particle executing simple harmonic motion is given by

x (t) = A sin (t + )

where

A is the maximum displacement of the particle from equilibrium position called amplitude.

is the angular velocity of imaginary reference particle in circular motion whose projection on the line is given simple harmonic motion, and is called angular frequency and

is the initial phase angle of the reference particle called phase constant.

Comparing the given equation with the equation of simple harmonic motion we can make conclusions

The motion of the raft is simple harmonic with amplitude of A = 0.60 m, angular frequency of simple harmonic motion = 3 radians/s and initial phase or phase constant is 4.2 rad or 4.2 degree.

1. What is the boat's position at t=0? At t=2.8s?

The position of the raft at t = 0 is given by

X(0) = 0.60 sin (3.0*0 rad + 4.2 rad) = 0.60* sin (4.2 rad)

= 0.60*sin = 0.60*sin 240.60

= 0.60*(-0.87) = - 0.523 m

Or X(0) = 0.60 sin (3.0*0 rad + 4.20) = 0.60* sin

Or x(0) = 0.60*0.073 = 0.044 m

The position of the raft at t = 2.8s is given by

X(2.8s) = 0.60 sin (3.0*2.8 rad + 4.2 rad) = 0.60* sin (12.6 rad)

= 0.60*sin = 0.60*sin 721.930

= 0.60*(0.0336) = 0.020 m

Or X(0) = 0.60 sin (3.0*2.8 rad + 4.20)

= 0.60* sin (3.0*2.8*57.29 +4.2)0

= 0.60*0.814 = 0.488 m

2. What is the period of the raft's oscillation?

The period of oscillation is given by

s

3. What is the boat's maximum speed during this oscillation?

The velocity v of a particle executing simple harmonic motion as a function of time t is given by

Or

The value of this function will be maximum for the value of is maximum and that is one hence the maximum speed of the raft is given by

m/s

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