A 2.10-kg object is attached to a spring and placed on frictionless, horizontal

Question

A 2.10-kg object is attached to a spring and placed on frictionless, horizontal surface. A horizontal force of 21.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest from this stretched position, and it subsequently undergoes simple harmonic oscillations. (a) Find the force constant of the spring N/m (b) Find the frequency of the oscillations Hz (c) Find the maximum speed of the object. m/s (d) Where does this maximum speed occur? (e) Find the maximum acceleration of the object. m/s2 (f) Where does the maximum acceleration occur? (9) Find the total energy of the oscillating system. (h) Find the speed of the object when its position is equal to one-third of the maximum value m/s (i) Find the magnitude of the acceleration of the object when its position is equal to one-third of the maximum value m/s2

Explanation / Answer

A.

Restoring force on spring

F= - kx

k= -F/x

k= 21/0.2

k= 105 N/m

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B.

Frequency is given by

f= (1/2*pi)*(k/m)^0.5

f= (1/2*3.14)* (105/2.1)^0.5

= 1.126 Hz

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C.

Velocity,

v= (k/m)^0.5* x

v= 1.414 m/s

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D.

Max speed observed at mean position.

x= + - 0

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E.

Max acceleration

a= kx/m

= 105* 0.2/2.1

= 10 m/s^2

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F.

Acceleration is max at Max displacement

a= + - 0.2 m

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g.

E= 0.5* kx^2

= 0.5* 105* 0.2*0.2

= 2.1 J

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H.

Velocity is given as

V= (k(A^2-x^2)/m)^0.5

=1.33 m/s

------

I

a= k x'/ m

= 105* 0.2/(3* 2.1)

= 3.33 m/s^2

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Comment in case any doubt . good luck

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