(a) 0.033 m
There are two forces acting on the box as it touches the spring:
- Gravity: [tex]W=mg[/tex], downward, with m=1.5 kg being the mass of the box and g=9.8 m/s^2 being the gravitational acceleration
- The force of the spring: [tex]F=kx[/tex], upward, with k=450 N/m being the spring constant x being the displacement of the spring
According to Newton's second law, we have:
[tex]mg-kx=ma[/tex]
when the acceleration of the box is zero, a=0, so the previous equation becomes
[tex]mg-kx=0[/tex]
And solving for x we find the displacement of the spring when the box's acceleration is zero:
[tex]k=\frac{mg}{k}=\frac{(1.5 kg)(9.8 m/s^2)}{450 N/m}=0.033 m=3.3 cm[/tex]
(b) 0.028 m
When the spring is fully compressed, the box's speed is zero, so this means that the whole initial kinetic energy of the box has been converted into elastic potential energy of the spring:
[tex]\frac{1}{2}mv^2 = \frac{1}{2}kx^2[/tex]
where v=0.49 m/s is the initial speed of the spring. Solving for x, we find the compression of the spring:
[tex]x=\sqrt{\frac{mv^2}{k}}=\sqrt{\frac{(1.5 kg)(0.49 m/s)^2}{450 N/m}}=0.028 m=2.8 cm[/tex]
The magnitude of the spring's displacement when the spring is fully compressed is 0.076m.
Spring constant = 450N/m
Mass = 1.5kg
At equilibrium, spring force will be equal to the weight of block. Therefore, kx = mg.
450(x) = 1.5 × 9.8
x = (1.5 ×9.8) / 450
x = 0.0327m
The magnitude of the spring's displacement when the spring is fully compressed will be:
0.5 × mv² + (1.5 × 9.8) = (0.5 × 450)
0.5 × 1.5 × 0.49² + (1.5 × 9.8) = (0.5 × 450
= 0.076m
Therefore, the magnitude is 0.076m.
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