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A spring is stretched 157 mm by a 7-kg block. If the block is displaced 91 mm downward from its equilibrium position and given a downward velocity of 1.2 m/s, determine the position of the block when [tex]t = 0.22[/tex] s.

Report your answer in meters, not millimeters. Use three decimal places in your answer.

Answer :

If its equilibrium position and given a downward velocity of 1.2 m/s, determine the position of the block when t =0.22 s then the position of the block when t = 0.22 s is 0.046 m.

How to find?

Using the equation of motion for simple harmonic motion, we can determine the position of the block as follows:

x = [tex]Acos(ωt + ϕ) + x₀[/tex]

Where;

x = position of the block

A = amplitude

ω = angular frequency

t = time elapsedϕ

= phase constant

x₀ = initial displacement

Also, since it is displaced 91 mm downward from its equilibrium position, its initial displacement is

x₀ = -91 mm

= -0.091 m.

Hence, substituting all these values in the formula above;

x = [tex]Acos(ωt + ϕ) + x₀[/tex]

We have;

x = [tex]-0.091 cos(2.093t + ϕ) + 0.091.[/tex]

Taking the derivative of the position function, we can obtain the velocity function for the block as follows;

v = [tex]-Aω sin(ωt + ϕ).[/tex]

Therefore, we need to find the phase constant ϕ such that the displacement x and velocity v of the block at time t = 0.22 s is as follows;

x = -0.091 cos(2.093t + ϕ) + 0.091

= -0.091 cos(2.093 × 0.22 + ϕ) + 0.091

= -0.0576 + 0.091 cos ϕv

= 0.191 sin(2.093t + ϕ)

= 0.191 sin(2.093 × 0.22 + ϕ)

= 0.128 sin ϕAt time t = 0.22 s,

the velocity of the block is given as 1.2 m/s, and it is moving downwards.

Therefore;

1.2 = 0.191 sin ϕ.

Since sin ϕ can take only values between -1 and 1, the possible values of ϕ are as follows;

ϕ₁ = sin⁻¹(1.2/0.191)

= 1.270 radϕ₂

= π - sin⁻¹(1.2/0.191)

= 1.871 rad.

We can then find the displacement of the block when t = 0.22 s as follows;

x = -0.091 cos(2.093t + ϕ) + 0.091

= -0.091 cos(2.093 × 0.22 + 1.871) + 0.091

= 0.046 m.

Therefore, the position of the block when t = 0.22 s is 0.046 m.

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