a mass is connected to a spring . if it is displaced a little from the rest position it oscilates where the position is given by x= A*sin (wt) where A and W are constants . find the velocity v(t)
Answers
Explanation:
oscillates in one dimension with the force of the spring acting parallel to the motion:
W
=
x
f
∫
x
i
F
x
d
x
=
x
f
∫
x
i
−
k
x
d
x
=
[
−
1
2
k
x
2
]
x
f
x
i
=
−
[
1
2
k
x
2
f
−
1
2
k
x
2
i
]
=
−
[
U
f
−
U
i
]
=
−
Δ
U
.
When considering the energy stored in a spring, the equilibrium position, marked as
x
i
=
0.00
m,
is the position at which the energy stored in the spring is equal to zero. When the spring is stretched or compressed a distance x, the potential energy stored in the spring is
U
=
1
2
k
x
2
.
Energy and the Simple Harmonic Oscillator
To study the energy of a simple harmonic oscillator, we need to consider all the forms of energy. Consider the example of a block attached to a spring, placed on a frictionless surface, oscillating in SHM. The potential energy stored in the deformation of the spring is
U
=
1
2
k
x
2
.
In a simple harmonic oscillator, the energy oscillates between kinetic energy of the mass
K
=
1
2
m
v
2
and potential energy
U
=
1
2
k
x
2
stored in the spring. In the