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Question Number 224853 by fantastic last updated on 07/Oct/25
a piece of chalk rests on a horizontal board with μ=0.1 Suddenly the board starts to move horizontally at a speed of 2m per second and after a time τ it stops abruptly. find the length of the line drawn by the chalk on the board for folowing cases τ=5sec τ=1sec g=10m/s^2
$$ \\ $$$$\mathrm{a}\:\mathrm{piece}\:\mathrm{of}\:\mathrm{chalk}\:\mathrm{rests}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{board}\:\mathrm{with}\:\mu=\mathrm{0}.\mathrm{1} \\ $$$$\mathrm{Suddenly}\:\mathrm{the}\:\mathrm{board}\:\mathrm{starts}\:\mathrm{to} \\ $$$$\mathrm{move}\:\mathrm{horizontally}\:\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{2m}\:\mathrm{per}\:\mathrm{second}\:\mathrm{and}\:\mathrm{after}\:\mathrm{a} \\ $$$$\mathrm{time}\:\tau\:\mathrm{it}\:\mathrm{stops}\:\mathrm{abruptly}.\:\mathrm{find}\: \\ $$$$\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{drawn} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{chalk}\:\mathrm{on}\:\mathrm{the}\:\mathrm{board}\:\mathrm{for} \\ $$$$\mathrm{folowing}\:\mathrm{cases} \\ $$$$\tau=\mathrm{5}{sec} \\ $$$$\tau=\mathrm{1}{sec} \\ $$$${g}=\mathrm{10}{m}/{s}^{\mathrm{2}} \\ $$
Commented by mr W last updated on 14/Oct/25
do you have the answers?
$${do}\:{you}\:{have}\:{the}\:{answers}? \\ $$
Commented by fantastic last updated on 14/Oct/25
yes
$${yes} \\ $$
Commented by fantastic last updated on 14/Oct/25
i posted it here because i was f acing problem with the voice of the teacher. if you know Hindi you will face no problem. other wise you can turn on subtitel. I can simply tell you the solution but I highly recommend you and other iterested people to watch the video . the Q needs a very high visualisation.
$${i}\:{posted}\:{it}\:{here}\:{because}\:{i}\:{was} \\ $$$${f}\:{acing}\:\:{problem}\:{with}\:{the}\:{voice} \\ $$$${of}\:{the}\:{teacher}.\:{if}\:{you}\:{know}\:{Hindi} \\ $$$${you}\:{will}\:{face}\:{no}\:{problem}.\:{other}\:{wise}\: \\ $$$${you}\:{can}\:{turn}\:{on}\:{subtitel}. \\ $$$${I}\:{can}\:{simply}\:{tell}\:{you}\:{the}\:{solution} \\ $$$${but}\:{I}\:{highly}\:{recommend}\:{you}\:{and} \\ $$$$\:{other}\:{iterested}\:{people}\:{to} \\ $$$${watch}\:{the}\:{video}\:. \\ $$$${the}\:{Q}\:{needs}\:{a}\:{very}\:{high}\:{visualisation}. \\ $$
Commented by fantastic last updated on 14/Oct/25
a) 2m b) 1.5m
$$\left.{a}\right)\:\mathrm{2}{m} \\ $$$$\left.{b}\right)\:\mathrm{1}.\mathrm{5}{m} \\ $$
Commented by fantastic last updated on 14/Oct/25
https://youtu.be/sL4vvgR4Alg?si=ujzpEG4Q9LIVsrby
Commented by mr W last updated on 14/Oct/25
i got the same results. thanks! i try to solve the questions in this forum purely for fun. usually i try to solve them in my own way and am not really interested how other people solve. i′m from Germany and don′t understand Hindi either.
$${i}\:{got}\:{the}\:{same}\:{results}.\:{thanks}! \\ $$$${i}\:{try}\:{to}\:{solve}\:{the}\:{questions}\:{in}\:{this} \\ $$$${forum}\:{purely}\:{for}\:{fun}.\:{usually}\:{i}\:{try} \\ $$$${to}\:{solve}\:{them}\:{in}\:{my}\:{own}\:{way}\:{and}\:{am} \\ $$$${not}\:{really}\:{interested}\:{how}\:{other} \\ $$$${people}\:{solve}. \\ $$$${i}'{m}\:{from}\:{Germany}\:{and}\:{don}'{t} \\ $$$${understand}\:{Hindi}\:{either}. \\ $$
Commented by fantastic last updated on 14/Oct/25
wow!
$${wow}!\: \\ $$
Commented by fantastic last updated on 14/Oct/25
Sir i think you mingled the 1st and 2nd case
$${Sir}\:{i}\:{think}\:{you}\:{mingled} \\ $$$${the}\:\mathrm{1}{st}\:{and}\:\mathrm{2}{nd}\:{case} \\ $$
Commented by fantastic last updated on 14/Oct/25
only one Q. why the f_(kinetic) will act at the same direction of the boards velocity?
$${only}\:{one}\:{Q}. \\ $$$${why}\:{the}\:{f}_{{kinetic}} \:{will}\:{act}\:{at}\:{the} \\ $$$${same}\:{direction}\:{of}\:{the}\:{boards}\:{velocity}? \\ $$$$ \\ $$
Commented by mr W last updated on 15/Oct/25
the question asked two cases. it′s inessential which case is treated as first and which as second. it′s unfortunate that you pointed this out as if it were a fault.
$${the}\:{question}\:{asked}\:{two}\:{cases}.\:{it}'{s} \\ $$$${inessential}\:{which}\:{case}\:{is}\:{treated} \\ $$$${as}\:{first}\:{and}\:{which}\:{as}\:{second}. \\ $$$${it}'{s}\:{unfortunate}\:{that}\:{you}\:{pointed}\:{this} \\ $$$${out}\:{as}\:{if}\:{it}\:{were}\:{a}\:{fault}. \\ $$
Commented by mr W last updated on 15/Oct/25
Commented by mr W last updated on 15/Oct/25
but at the moment when the board stops abruptly, the board rests while the chalk continues to move.
$${but}\:{at}\:{the}\:{moment}\:{when}\:{the}\:{board} \\ $$$${stops}\:{abruptly},\:{the}\:{board}\:{rests}\:{while} \\ $$$${the}\:{chalk}\:{continues}\:{to}\:{move}. \\ $$
Commented by mr W last updated on 15/Oct/25
Kinetic friction tries to reduce the relative motion between two objects. at t=0, the board moves to right, say, and the chalk remains in rest. the direction of friction is always opposite to the direction of the relative motion of the objects.
Kinetic friction tries to reduce the relative motion between two objects.
$${at}\:{t}=\mathrm{0},\:{the}\:{board}\:{moves}\:{to}\:{right},\:{say}, \\ $$$${and}\:{the}\:{chalk}\:{remains}\:{in}\:{rest}.\:{the} \\ $$$${direction}\:{of}\:{friction}\:{is}\:{always}\: \\ $$$${opposite}\:{to}\:{the}\:{direction}\:{of}\:{the} \\ $$$${relative}\:{motion}\:{of}\:{the}\:{objects}. \\ $$
Commented by mr W last updated on 15/Oct/25
Commented by fantastic last updated on 15/Oct/25
thank you sir
$${thank}\:{you}\:{sir} \\ $$
Answered by mr W last updated on 16/Oct/25
at t=0: both board and chalk are in rest. when the board begins to move with speed v=2 m/s suddently, say towards right, the challk remains in rest, but obtains a kinetic friction force f in the direction of the movement of the board. this friction force brings the chalk to move with an acceleration a. ma=f=μmg ⇒a=μg=0.1×10=1 m/s^2 after time t, the distance the board has moved is s_B =vt=2t and the distance the chalk has moved is s_C =((at^2 )/2)=0.5t^2 as soon as the chalk has the same velocity as the board, there is no relative motion between them and therefore no friction. upon now both object move with the same velocity. at=v ⇒t=(v/a)=(2/1)=2 s. i.e. after 2 s. the length of the line drawn by the chalk on the board is the relative displacement between both objects: d=s_B −s_C =2t−0.5t^2
$${at}\:{t}=\mathrm{0}: \\ $$$${both}\:{board}\:{and}\:{chalk}\:{are}\:{in}\:{rest}. \\ $$$${when}\:{the}\:{board}\:{begins}\:{to}\:{move}\:{with} \\ $$$${speed}\:{v}=\mathrm{2}\:{m}/{s}\:{suddently},\:{say}\: \\ $$$${towards}\:{right},\:{the}\:{challk}\:{remains}\: \\ $$$${in}\:{rest},\:{but}\:{obtains}\:{a}\:{kinetic}\:{friction} \\ $$$${force}\:{f}\:{in}\:{the}\:{direction}\:{of}\:{the}\: \\ $$$${movement}\:{of}\:{the}\:{board}.\:{this}\:{friction} \\ $$$${force}\:{brings}\:{the}\:{chalk}\:{to}\:{move}\:{with} \\ $$$${an}\:{acceleration}\:{a}. \\ $$$${ma}={f}=\mu{mg}\: \\ $$$$\Rightarrow{a}=\mu{g}=\mathrm{0}.\mathrm{1}×\mathrm{10}=\mathrm{1}\:{m}/{s}^{\mathrm{2}} \\ $$$${after}\:{time}\:{t},\:{the}\:{distance}\:{the}\:{board} \\ $$$${has}\:{moved}\:{is} \\ $$$${s}_{{B}} ={vt}=\mathrm{2}{t} \\ $$$${and}\:{the}\:{distance}\:{the}\:{chalk}\:{has}\:{moved} \\ $$$${is} \\ $$$${s}_{{C}} =\frac{{at}^{\mathrm{2}} }{\mathrm{2}}=\mathrm{0}.\mathrm{5}{t}^{\mathrm{2}} \\ $$$${as}\:{soon}\:{as}\:{the}\:{chalk}\:{has}\:{the}\:{same} \\ $$$${velocity}\:{as}\:{the}\:{board},\:{there}\:{is}\:{no} \\ $$$${relative}\:{motion}\:{between}\:{them}\:{and} \\ $$$${therefore}\:{no}\:{friction}.\:{upon}\:{now} \\ $$$${both}\:{object}\:{move}\:{with}\:{the}\:{same} \\ $$$${velocity}.\: \\ $$$${at}={v}\:\Rightarrow{t}=\frac{{v}}{{a}}=\frac{\mathrm{2}}{\mathrm{1}}=\mathrm{2}\:{s}. \\ $$$${i}.{e}.\:{after}\:\mathrm{2}\:{s}. \\ $$$${the}\:{length}\:{of}\:{the}\:{line}\:{drawn}\:{by}\:{the} \\ $$$${chalk}\:{on}\:{the}\:{board}\:{is}\:{the}\:{relative} \\ $$$${displacement}\:{between}\:{both}\:{objects}: \\ $$$${d}={s}_{{B}} −{s}_{{C}} =\mathrm{2}{t}−\mathrm{0}.\mathrm{5}{t}^{\mathrm{2}} \\ $$
Commented by mr W last updated on 13/Oct/25
Commented by mr W last updated on 13/Oct/25
the trace of the chalk on the board:
$${the}\:{trace}\:{of}\:{the}\:{chalk}\:{on}\:{the}\:{board}: \\ $$
Commented by mr W last updated on 14/Oct/25
case 1: the board stops abruptly after 1 s. at t=1 s: distance moved by the board is s_B =2×1=2 m distance moved by the chalk is s_C =0.5×1^2 =0.5 m at this moment the velocity of the chalk is 1×1=1 m/s. after this moment the board is kept in rest, but the chalk remains in motion. but the friction is now in opposite direction. that means the acceleration is also in opposite direction: a=−μg=−1 m/s^2 after 1 s the chalk also stops. during this time the chalk moves further 0.5 m. the maximal relative displacement and therefore the length of the line drawn by the chalk is at t=1 s: s_B −s_C =2−0.5=1.5 m.
$${case}\:\mathrm{1}:\: \\ $$$${the}\:{board}\:{stops}\:{abruptly}\:{after}\:\mathrm{1}\:{s}. \\ $$$${at}\:{t}=\mathrm{1}\:{s}:\: \\ $$$${distance}\:{moved}\:{by}\:{the}\:{board}\:{is}\: \\ $$$${s}_{{B}} =\mathrm{2}×\mathrm{1}=\mathrm{2}\:{m} \\ $$$${distance}\:{moved}\:{by}\:{the}\:{chalk}\:{is}\: \\ $$$${s}_{{C}} =\mathrm{0}.\mathrm{5}×\mathrm{1}^{\mathrm{2}} =\mathrm{0}.\mathrm{5}\:{m} \\ $$$${at}\:{this}\:{moment}\:{the}\:{velocity}\:{of}\:{the} \\ $$$${chalk}\:{is}\:\mathrm{1}×\mathrm{1}=\mathrm{1}\:{m}/{s}. \\ $$$${after}\:{this}\:{moment}\:{the}\:{board}\:{is} \\ $$$${kept}\:{in}\:{rest},\:{but}\:{the}\:{chalk}\:{remains} \\ $$$${in}\:{motion}.\:{but}\:{the}\:{friction}\:{is}\:{now}\:{in} \\ $$$${opposite}\:{direction}.\:{that}\:{means}\:{the} \\ $$$${acceleration}\:{is}\:{also}\:{in}\:{opposite}\: \\ $$$${direction}: \\ $$$${a}=−\mu{g}=−\mathrm{1}\:{m}/{s}^{\mathrm{2}} \\ $$$${after}\:\mathrm{1}\:{s}\:{the}\:{chalk}\:{also}\:{stops}.\:{during} \\ $$$${this}\:{time}\:{the}\:{chalk}\:{moves}\:{further} \\ $$$$\mathrm{0}.\mathrm{5}\:{m}. \\ $$$${the}\:{maximal}\:{relative}\:{displacement} \\ $$$${and}\:{therefore}\:{the}\:{length}\:{of}\:{the} \\ $$$${line}\:{drawn}\:{by}\:{the}\:{chalk}\:{is}\:{at}\:{t}=\mathrm{1}\:{s}: \\ $$$${s}_{{B}} −{s}_{{C}} =\mathrm{2}−\mathrm{0}.\mathrm{5}=\mathrm{1}.\mathrm{5}\:{m}. \\ $$
Commented by mr W last updated on 14/Oct/25
Commented by mr W last updated on 13/Oct/25
Commented by mr W last updated on 13/Oct/25
the trace of the chalk on the board:
$${the}\:{trace}\:{of}\:{the}\:{chalk}\:{on}\:{the}\:{board}: \\ $$
Commented by mr W last updated on 14/Oct/25
case 2: the board stops abruptly after 5 s. at t=2 s: velocity of chalk is 1×2=2 m/s. distance moved by board is 2×2=4 m. distance moved by chalk is 0.5×1×2^2 =2 m. the relative displacement is 4−2=2 m. after this moment the chalk moves with the same speed as the board, so the friction force and therefore also its acceleration is zero. at t=5 s: the board stops abruptly. after this moment the board is kept in rest, but the chalk remains in motion and with an acceleration in opposite direction a=−μg=−1 m/s^2 . after 2 s also the chalk stops. during this time the distance moved by the chalk is 2×2−0.5×1×2^2 =2 m. the length of the line drawn by the chalk is the maximal relative displacement at t=2 till 5 s: s_B −s_C =4−2=2 m.
$${case}\:\mathrm{2}:\: \\ $$$${the}\:{board}\:{stops}\:{abruptly}\:{after}\:\mathrm{5}\:{s}. \\ $$$${at}\:{t}=\mathrm{2}\:{s}:\: \\ $$$${velocity}\:{of}\:{chalk}\:{is}\:\mathrm{1}×\mathrm{2}=\mathrm{2}\:{m}/{s}. \\ $$$${distance}\:{moved}\:{by}\:{board}\:{is}\:\mathrm{2}×\mathrm{2}=\mathrm{4}\:{m}. \\ $$$${distance}\:{moved}\:{by}\:{chalk}\:{is}\:\mathrm{0}.\mathrm{5}×\mathrm{1}×\mathrm{2}^{\mathrm{2}} =\mathrm{2}\:{m}. \\ $$$${the}\:{relative}\:{displacement}\:{is}\:\mathrm{4}−\mathrm{2}=\mathrm{2}\:{m}. \\ $$$${after}\:{this}\:{moment}\:{the}\:{chalk}\:{moves} \\ $$$${with}\:{the}\:{same}\:{speed}\:{as}\:{the}\:{board},\:{so} \\ $$$${the}\:{friction}\:{force}\:{and}\:{therefore}\: \\ $$$${also}\:{its}\:{acceleration}\:{is}\:{zero}. \\ $$$${at}\:{t}=\mathrm{5}\:{s}: \\ $$$${the}\:{board}\:{stops}\:{abruptly}. \\ $$$${after}\:{this}\:{moment}\:{the}\:{board}\:{is} \\ $$$${kept}\:{in}\:{rest},\:{but}\:{the}\:{chalk}\:{remains} \\ $$$${in}\:{motion}\:{and}\:{with}\:{an}\:{acceleration} \\ $$$${in}\:{opposite}\:{direction} \\ $$$${a}=−\mu{g}=−\mathrm{1}\:{m}/{s}^{\mathrm{2}} . \\ $$$${after}\:\mathrm{2}\:{s}\:{also}\:{the}\:{chalk}\:{stops}.\:{during} \\ $$$${this}\:{time}\:{the}\:{distance}\:{moved}\:{by}\:{the} \\ $$$${chalk}\:{is}\:\mathrm{2}×\mathrm{2}−\mathrm{0}.\mathrm{5}×\mathrm{1}×\mathrm{2}^{\mathrm{2}} =\mathrm{2}\:{m}. \\ $$$$ \\ $$$${the}\:{length}\:{of}\:{the}\:{line}\:{drawn}\:{by}\:{the} \\ $$$${chalk}\:{is}\:{the}\:{maximal}\:{relative} \\ $$$${displacement}\:{at}\:{t}=\mathrm{2}\:{till}\:\mathrm{5}\:{s}: \\ $$$${s}_{{B}} −{s}_{{C}} =\mathrm{4}−\mathrm{2}=\mathrm{2}\:{m}. \\ $$
Commented by mr W last updated on 14/Oct/25
Commented by mr W last updated on 14/Oct/25
Commented by ajfour last updated on 16/Oct/25
https://youtu.be/G_goL8NB2JQ?si=3Ybz8e5654HFqO4I

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