A 3.60-kg object is attached to a spring and placed on frictionless, horizontal

Question

A 3.60-kg object is attached to a spring and placed on frictionless, horizontal surface. A horizontal force of 17.0 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?
x =

Explanation / Answer

(a)

The force constant is F / x = 17 / 0.2

k = 85.0 N/m
the force constant is 85 N/m

(b)

f = (1/2?)sqrt(k/m)

f = (1/2?) sqrt( 85 / 3.6) = 0.772 Hz

(c)

The displacement is given by A*sin(2?ft)

A = initial deflection = 0.200. The velocity is the derivative of this or 2?fA*cos(2?ft).

(d)

At max (where cos = 1) the velocity is 2?fA

so

velocity = 2 *3.14 * 0.772 *0.2 = 0.97 m/s

(e)

The acceleration is the derivative of the velocity or -(2?f)^2A*sin(2?ft)

and the value of the max is (2?f)^2A = (2* 3.14 * 0.772)^2 * 0.2 = 4.7 m/s^2

(f)

The max acceleration occurs when the velocity is zero, which is when the mass passes through the neutral point of the spring. the postion of the string when it is unforced

(g)

The total energy is the same as the initial energy, which is 0.5*k*x^2 = 1.7 J
so the total energy is 1.7J

(h)

The position is A*sin(2?ft).

The time when the position is A/3 is given by
1/3 = sin(2?ft)

so

2?ft = arcsin(1/3). The velocity is 2?fA*cos(2?ft) so at A/3 v = 2?fA*cos[arcsin(1/3)]

so the speed off the object = 0.97 * 0.942 = 0.913 m/s

(h)

Acceleration is -(2?f)^2A*sin(2?ft) and

at A/3 2?ft = arcsin(1/3) sin[arcsin(1/3)] = 1/3 so the acceleration is -(2?f)^2A/3 = - 4.70/3 = 1.56 m/s^2
so the acceleration is 1.56 m/s^2

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