Problem $7$: $(7\%$ of Assignment Value$)$

A sphere of mass $m=3\mathrm{.}5kg$ can move across a horizontal, frictionless surface. Attached to the sphere is an ideal spring with spring constant $k=34\frac{N}{m}.$ At time $t=0$ the sphere is pulled aside from the equilibrium position, $x=0,$ a distance $d=12cm$ in the positive direction and released from rest. After this time, the system oscillates between $x=+-d.$

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Part $($a$)$

Determine the magnitude of the force, in newtons, required to initially displace the sphere $d=12$ centimeters from equilibrium.

$F=4\mathrm{.}080N,$ Correct!

Part $($b$)$

What is the sphere's distance from equilibrium, in meters, at time $t=1$ second?

$x(1)=$

$$ m

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Part $($c$)$

Determine the frequency, in hertz, with which the spring$-$mass system oscillates after being released.

$f=0\mathrm{.}4890Hz$

$$ Correct!

Part $($d$)$

Calculate the maximum speed, in meters per second, attained by the sphere.

$v_{\text{m}\text{a}\text{x}}=0\mathrm{.}3690\frac{m}{s}$

$$ Correct!

Part $($e$)$

At what point in the motion does the sphere reach maximum speed?

at $x=0$

$$ Correct!

Part $($f$)$

Calculate the magnitude of the maximum acceleration, in meters per second squared, experienced by the sphere.

$a_{\text{m}\text{a}\text{x}}=1\mathrm{.}130\frac{m}{s^2}$

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