(a) 0.456 m/s
The maximum speed of the oscillating mass can be found by using the conservation of energy. In fact:
- At the point of maximum displacement, the mechanical energy of the system is just elastic potential energy:
[tex]E=U=\frac{1}{2}kA^2[/tex] (1)
where
k = 34.9 N/m is the spring constant
A = 5.0 cm = 0.05 m is the amplitude of the oscillation
- At the point of equilibrium, the displacement is zero, so all the mechanical energy of the system is just kinetic energy:
[tex]E=K=\frac{1}{2}mv_{max}^2[/tex] (2)
where
m = 0.42 kg is the mass
vmax is the maximum speed, which is maximum when the mass passes the equilibrium position
Since the mechanical energy is conserved, we can write (1) = (2):
[tex]\frac{1}{2}kA^2=\frac{1}{2}mv_{max}^2\\v_{max}=\sqrt{\frac{kA^2}{m}}=\sqrt{\frac{(34.9 N/m)(0.05 m)^2}{0.42 kg}}=0.456 m/s[/tex]
(b) 0.437 m/s
When the spring is compressed by x = 1.5 cm = 0.015 m, the equation for the conservation of energy becomes:
[tex]E=\frac{1}{2}kx^2+\frac{1}{2}mv^2[/tex] (3)
where the total mechanical energy can be calculated at the point where the displacement is maximum (x = A = 0.05 m):
[tex]E=\frac{1}{2}kA^2=\frac{1}{2}(34.9 N/m)(0.05 m)^2=0.044 J[/tex]
So, solving (3) for v, we find the speed when x=1.5 cm:
[tex]v=\sqrt{\frac{2E-kx^2}{m}}=\sqrt{\frac{2(0.044 J)-(34.9 N/m)(0.015 m)^2}{0.42 kg}}=0.437 m/s[/tex]
(c) 0.437 m/s
This part of the problem is exactly identical to part b), since the displacement of the mass is still
x = 1.5 cm = 0.015 m
So, the speed when this is the displacement is
[tex]v=\sqrt{\frac{2E-kx^2}{m}}=\sqrt{\frac{2(0.044 J)-(34.9 N/m)(0.015 m)^2}{0.42 kg}}=0.437 m/s[/tex]
(d) 4.4 cm
In this case, we have that the speed of the mass is 1/2 of the maximum value, so:
[tex]v=\frac{v_{max}}{2}=\frac{0.456 m/s}{2}=0.228 m/s[/tex]
And by using the conservation of energy again, we can find the corresponding value of the displacement x:
[tex]E=\frac{1}{2}kx^2+\frac{1}{2}mv^2\\x=\sqrt{\frac{2E-mv^2}{k}}=\sqrt{\frac{2(0.044 J)-(0.42 kg)(0.228 m/s)^2}{34.9 N/m}}=0.044 m=4.4 cm[/tex]
