A particle moves along the x axis. It is initially at the position 0.140 m, movi

Question

A particle moves along the x axis. It is initially at the position 0.140 m, moving with velocity 0.150 m/s and acceleration-0.240 m/s2. Suppose it moves with constant acceleration for 3.70 s.

(a) Find the position of the particle after this time.___________ m

(b) Find its velocity at the end of this time interval.__________ m/s

We take the same particle and give it the same initial conditions as before. Instead of having a constant acceleration, it oscillates in simple harmonic motion for 3.70 s around the equilibrium position x = 0. Hint: the following problems are very sensitive to rounding, and you should keep all digits in your calculator.

(c) Find the angular frequency of the oscillation. Hint: in SHM, a is proportional to x._____ /s

(d) Find the amplitude of the oscillation. Hint: use conservation of energy.______ m

(e) Find its phase constant 0 if cosine is used for the equation of motion. Hint: when taking the inverse of a trig function, there are always two angles but your calculator will tell you only one and you must decide which of the two angles you need.____ rad

(f) Find its position after it oscillates for 3.70 s._______ m

(g) Find its velocity at the end of this 3.70 s time interval._______ m/s

Explanation / Answer

a) and b) are motion with constant linear acceleration

a)
x(t) = x(o) + v(o)×t + a×t²/2
x(3.7) = (0.14 m) + (0.15m/s)×(3.7 s) + (-0.24 m/s²)×(3.7s)²/2
x(3.7) = -0.9478 m

b)
v(t) = v(o) + a×t
v(3.7) = (0.14 m/s) + (-0.24 m/s²)×(3.7s)
v(3.7) = -0.748 m/s

c) through g) are simple harmonic motion (SHM)

The equation for position of SHM as a function of time is
x(t) = A cos(t + )

The equation for velocity of SHM as a function of time is:
v(t) = dx/dt = d(A cos(t + ))/dt
v(t) = -A sin(t + )

The equation for acceleration of SHM as a function of time is:
a(t) = dv/dt = d(-A sin(t + ))/dt
a(t) = -A ² cos(t + )

where
A = amplitude
= angular velocity (or angular frequency)
= angular phase shift

Nothing is given about position, velocity or acceleration AT A SPECIFIC TIME, so you can choose the time to be t=0 when the particle is moving through x=0.14
At that point you know that:
x(o) = A cos(×0 + ) = 0.14 m
v(o) = -A sin(×0 + ) = 0.15 m/s
a(o) = -A ² cos(×0 + ) = -0.24 m/s²
which simplifies to 3 equations with 3 unknowns:
A cos() = 0.14 m
A sin() = -0.15 m/s
A ² cos() = 0.24 m/s²

Solving this system of equations, you will find:
= 1.31 rad/s <- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - answer c
A = m <- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - answer d
= - rad <- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - answer e
(contact me if you don't know how to solve this system)

Filling in t=3.7 into the equation for position, you get:

x(3.7) = A cos(×t + ) = m  - answer f
v(3.7) = -A sin(×t + ) = m/s  - answer g

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