Question

Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position y = 0 when the mass hangs at rest. Suppose you push the mass to a position yo units above its equilibrium position and release it. As the mass oscillates up and down (neglecting air friction), the position y of the mass after t seconds is given by the equation below. Use this equation to answer the questions below:

y = yocos (t square root k/m)

a) Find dy/dx, the velocity of the mass. Assumie k and m are constant.
b) How would the velocity be affected if the experiment were repeated with four times the mass on the end of the spring?
c) How would the velocity be affected if the experiment were repeated with a spring that has 4 times the stiffness (if k is increased by a factor of 4)?
d) Assume that y has units of meters, t has units of seconds, m has units of kg and k has units of kg/s2. Show that the units of the velocity in part a) are consistent.

Answer 1

Answer:

Explanation:

y = y₀ cos$\sqrt{\frac{k}{m} }\times t$

a )

$\frac{dy}{dt}$ = - y₀ x $\sqrt{\frac{k}{m} }$ sin (  $\sqrt{\frac{k}{m} }\times t$  )  

b ) If m = 4m

$\frac{dy}{dt}$ = - y $\sqrt{\frac{k}{4m} }$ sin ( $\sqrt{\frac{k}{4m} }\times t$  )

Magnitude of velocity will be decreased .

c )

$\frac{dy}{dt}$ = - y $\sqrt{\frac{4k}{m} }$ sin ( $\sqrt{\frac{4k}{m} }\times t$  )

magnitude of velocity will be increased .

d )

velocity = - y₀ $\sqrt{\frac{k}{m} }$ sin( $\sqrt{\frac{k}{m} }\times t$  )

              =  L $\sqrt{\frac{ms^{-2}}{m} }$  X 0

               = L s⁻¹

= m /s

unit of velocity is consistent .