Question 334: Work, Energy and Power & Circular Motion

Question

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $$2.0\,N{m^{ - 1}}$$ and the mass of the block is $$2.0$$ $$kg.$$ Ignore the mass of the spring. Initially the spring is an unstretched condition. Another block of mass $$1.0$$ $$kg$$ moving with a speed of $$2.0$$ $$m{s^{ - 1}}$$ collides elastically with the first block. The collision is  such that the $$2.0$$ $$kg$$ block does not hit the wall. The distance,  in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is __________.

Accepted Answer

After collision the 2.0 kg block will perform simple harmonic oscillation with time period
$$T = 2\pi \sqrt {{m \over k}} $$
Given m = 2.0 kg and k = 2.0 N m
$$-$$1
, we have
T = 2$$\pi$$
Thus, the block returns to its original position in time
$$t = {T \over 2}$$ = $$\pi$$ s
That is, t = 3.14 s
Now, if v
1
and v
2
are velocities of 1.0 kg block and 2.0 kg block, respectively, before collision; v'
1
and v'
2
are velocities of 1.0 kg block and 2.0 block, respectively, after collision. So, by conservation of momentum
m
1
v
1
+ m
2
v
2
= m
1
v'
1
+ m
2
v'
2
Here, m
1
= 1.0 kg, v
1
is initial speed of 1.0 kg block = 2.0 m s
$$-$$1
, m
2
= 2.0 kg, v
2
is initial speed of 2.0 kg block = 0.0 m s
$$-$$1
, v'
1
is final speed of 1.0 kg block after collision and v'
2
is final speed of 2.0 kg block after collision. Then, 1.0 kg $$	imes$$ 2.0 m/s + 2.0 kg $$	imes$$ 0 m/s = 1.0 v'
1
+ 2.0 v'
2
v'
1
+ 2v'
2
= 2 ..... (1)
Also, using definition of coefficient of restitution
v'
2
$$-$$ v'
1
= $$\varepsilon $$(v
1
$$-$$ v
2
)
Since collision is elastic, So $$\varepsilon $$ = 1
$$\Rightarrow$$ v'
2
$$-$$ v'
1
= v
1
$$-$$ v
2
$$\Rightarrow$$ v'
2
$$-$$ v'
1
= 2 $$-$$ 0
$$\Rightarrow$$ v'
2
$$-$$ v'
1
= 2 ...... (2)
From Eqs. (1) and (2), we get
$$v{'_2} = {4 \over 3}$$ m s
$$-$$1
and $$v{'_1} = {-2 \over 3}$$ m s
$$-$$1
Therefore, distance between the blocks is given as
$$s = v{'_1} 	imes t = {{ - 2} \over 3} 	imes 3.14 = 2.09$$ m

Work, Energy and Power & Circular Motion

Integer / Numeric Answer Type