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Question Number 223720 by fantastic last updated on 03/Aug/25
One end of a string is attached to a solid wall and the other end is hanging from a smooth pulley 2 m away fromthe wall. A point mass M of mass 2 kg is attached to the string 1 m away from the wall and an object m of mass 0.5 kg is attached to the hanging end of the string. The object is fixed in such a way that the part of the string inside the wall and pully is horizontal and the rest is vertical. If mass m is released, with what speed will mass M hit the wall?
$$ \\ $$$$\mathrm{One}\:\mathrm{end}\:\mathrm{of}\:\mathrm{a}\:\mathrm{string}\:\mathrm{is} \\ $$$$\mathrm{attached}\:\mathrm{to}\:\mathrm{a}\:\mathrm{solid}\:\mathrm{wall}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{other}\:\mathrm{end}\:\mathrm{is}\:\mathrm{hanging}\:\mathrm{from}\:\mathrm{a} \\ $$$$\mathrm{smooth}\:\mathrm{pulley}\:\mathrm{2}\:\mathrm{m}\:\mathrm{away}\: \\ $$$$\mathrm{fromthe}\:\mathrm{wall}.\:\mathrm{A}\:\mathrm{point}\:\mathrm{mass}\: \\ $$$$\mathrm{M}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}\:\mathrm{kg}\:\mathrm{is}\:\mathrm{attached}\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{string}\:\mathrm{1}\:\mathrm{m}\:\mathrm{away}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{wall}\:\mathrm{and}\:\mathrm{an}\:\mathrm{object}\:\mathrm{m}\:\mathrm{of}\:\mathrm{mass} \\ $$$$\mathrm{0}.\mathrm{5}\:\mathrm{kg}\:\mathrm{is}\:\mathrm{attached}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{hanging}\:\mathrm{end}\:\mathrm{of}\:\mathrm{the}\:\mathrm{string}.\: \\ $$$$\mathrm{The}\:\mathrm{object}\:\mathrm{is}\:\mathrm{fixed}\:\mathrm{in}\:\mathrm{such}\:\mathrm{a} \\ $$$$\mathrm{way}\:\mathrm{that}\:\mathrm{the}\:\mathrm{part}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{string}\:\mathrm{inside}\:\mathrm{the}\:\mathrm{wall}\:\mathrm{and} \\ $$$$\mathrm{pully}\:\mathrm{is}\:\mathrm{horizontal}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{rest}\:\mathrm{is}\:\mathrm{vertical}.\:\mathrm{If}\:\mathrm{mass}\:\mathrm{m}\:\mathrm{is} \\ $$$$\mathrm{released},\:\mathrm{with}\:\mathrm{what}\:\mathrm{speed}\:\mathrm{will} \\ $$$$\mathrm{mass}\:\mathrm{M}\:\mathrm{hit}\:\mathrm{the}\:\mathrm{wall}? \\ $$
Commented by fantastic last updated on 03/Aug/25
Answered by mr W last updated on 03/Aug/25
Commented by mr W last updated on 03/Aug/25
u=velocity of mass M v=velocity of mass m v=((2u)/( (√5))) h=(√5)−1 ((Mu^2 )/2)+((mv^2 )/2)=Mg×1−mg×h u^2 =((5(5−(√5))g)/(12)) ⇒u=(√((5(5−(√5))×9.81)/(12)))≈3.361 m/s
$${u}={velocity}\:{of}\:{mass}\:{M} \\ $$$${v}={velocity}\:{of}\:{mass}\:{m} \\ $$$${v}=\frac{\mathrm{2}{u}}{\:\sqrt{\mathrm{5}}} \\ $$$${h}=\sqrt{\mathrm{5}}−\mathrm{1} \\ $$$$\frac{{Mu}^{\mathrm{2}} }{\mathrm{2}}+\frac{{mv}^{\mathrm{2}} }{\mathrm{2}}={Mg}×\mathrm{1}−{mg}×{h} \\ $$$${u}^{\mathrm{2}} =\frac{\mathrm{5}\left(\mathrm{5}−\sqrt{\mathrm{5}}\right){g}}{\mathrm{12}} \\ $$$$\Rightarrow{u}=\sqrt{\frac{\mathrm{5}\left(\mathrm{5}−\sqrt{\mathrm{5}}\right)×\mathrm{9}.\mathrm{81}}{\mathrm{12}}}\approx\mathrm{3}.\mathrm{361}\:{m}/{s} \\ $$
Commented by fantastic last updated on 03/Aug/25
sir the answer is v≈11.3 m/s
$${sir}\:{the}\:{answer}\:{is}\:{v}\approx\mathrm{11}.\mathrm{3}\:{m}/{s} \\ $$
Commented by mr W last updated on 03/Aug/25
v≈11.3 m/s is impossible. even in free fall from 1 m height we have only a speed of (√(2×1×9.81))≈4.43 m/s. that means in current case the speed of M must be less than 4.43 m/s.
$${v}\approx\mathrm{11}.\mathrm{3}\:{m}/{s}\:{is}\:{impossible}.\:{even}\:{in} \\ $$$${free}\:{fall}\:{from}\:\mathrm{1}\:{m}\:{height}\:{we}\:{have} \\ $$$${only}\:{a}\:{speed}\:{of}\:\sqrt{\mathrm{2}×\mathrm{1}×\mathrm{9}.\mathrm{81}}\approx\mathrm{4}.\mathrm{43}\:{m}/{s}. \\ $$$${that}\:{means}\:{in}\:{current}\:{case}\:{the} \\ $$$${speed}\:{of}\:{M}\:{must}\:{be}\:{less}\:{than}\:\mathrm{4}.\mathrm{43}\:{m}/{s}. \\ $$
Commented by fantastic last updated on 03/Aug/25
I will send you the solution. It is done in my textbook but I dont understand it. or maybey I translated the question wrong
$${I}\:{will}\:{send}\:{you}\:{the}\:{solution}. \\ $$$${It}\:{is}\:{done}\:{in}\:{my}\:{textbook}\:{but}\:{I}\: \\ $$$${dont}\:{understand}\:{it}.\:{or}\:{maybey}\:{I} \\ $$$${translated}\:{the}\:{question}\:{wrong} \\ $$

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