Question

The solving to this question

Answer 1

This is a simple harmonic motion exercise. The proof that the resultant force on the particle, at time t, is -225 Newtons is indicated below.

What is the proof that the resultant force on the particle, at time t, is -225x newtons?

Note that we have been given the following:

Mass = 9kg,

Y = 45 N

ζ = 0.4 m

Then, Y/ζ  = 45N/0.4m

= 112.5 N/m

Hence, the spring constant =

K = 112. 5N/m

From this point we can resolve the Amplitude (A):
Amplitude is given as:

A = CB - PB

= 0.6 - 0.5

A = 0.1m

I) So because one spring is elongated by x and the other is compressed by x, Hence, the spring force is:

F$_{net}$ = - Kₓ - Kₓ

= - 2Kₓ
= -2 x 112.5 x* X

F$_{net}$ = -225x Newton ..........Lets call this expression 1

What is the proof that the particles move with simple harmonic motion?

From equation 1 above, we know that

Ma = -225x

a = (-225/m) x

⇒ a ∝ - x,

Hence, the motion is is simple harmonic in nature.

What is the period of the motion?

The time period, T = 2 π√m/K$_{eq}$

Where, K$_{eq}$ = 2K

= 2π√(9/225)

= 2π * 3/15

= 2(22/7) * (3/15)

= 1.25714285714; thus

T ≈ 1.26 secs...............lets call this expression 2

What is the speed of the particle when it is 0.05 meters from C?

Note that Angular Frequency is = 2π/T

= 2π/1.26

= 4.9866550057

ω = 4.99 rad/s

Since the velocity at any point x from the mean position is given by

V = ω √ (A² - x²); hence

V = 4.99 √[(0.1)² - (0.05)²

V = 4.99 x 0.0866
V = 0.432134

V ≈ 0.432..........Lets call this expression 3

What is the expression of x in terms of t?

Along the x-axis, A simple harmonic motion can be depicted as:

x = A sin (ωt + Ф) ..........lets call this expression 4

where, Ф = Phase constant

At t = 0; x = A

A = A sinФ

⇒ SinФ = 1

⇒ Ф = π/2

Expression 4 becomes:

x = A sin (ωt + π/2)

Inserting the values of ω and A, we have

x = 0.1 sin (4.99t + π/2).........Lets call this expression 5a

or  x  = 0.1 cos (4.99t).......Lets call this expression 5b

Hence,

Sin(π/2  + Ф) = CosФ

Learn more about simple harmonic motion at;
brainly.com/question/17315536
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