Question

a fan is rotating at 90 rpm.It is then switched off. It stops after 21 revolutions. calculate the time taken by the frictional torque is constant ​

Answer 1

ɪɴ ᴛʜɪs ᴄᴀsᴇ, ᴛʜᴇ ɪɴɪᴛɪᴀʟ ᴀɴɢᴜʟᴀʀ ᴠᴇʟᴏᴄɪᴛʏ ᴏғ ᴛʜᴇ ғᴀɴ ɪs ω =0

$</p><p>ω = 100 \: \: \: \: rev seconds</p><p>$

Aɴᴅ ᴛʜᴇ ғɪɴᴀʟ ᴀɴɢᴜʟᴀʀ ᴠᴇʟᴏᴄɪᴛʏ ᴏғ ᴛʜᴇ ғᴀɴ ɪs ω=100ʀᴇᴠ/sᴇᴄ.

ᴛʜᴇ ᴛᴏᴛᴀʟ ɴᴜᴍʙᴇʀ ᴏғ ʀᴇᴠᴏʟᴜᴛɪᴏɴs ʙᴇғᴏʀᴇ ᴛʜᴇ ғᴀɴ ᴄᴏᴍɪɴɢ ᴛᴏ sᴛᴏᴘ ɪs ᴄᴀʟᴄᴜʟᴀᴛᴇᴅ ᴀs ғᴏʟʟᴏᴡs.

$0 = (\frac{ \: \: </p><p>ω+ω </p><p> \: }{2} )t \\ = ( \frac{100 + 0}{2} ) \times (5 \times 60) \\ = 15000$

Answer 2

$\sf{\underline{\underline{\qquad \bigstar \: Given \: \bigstar\qquad }}}$

• Initial frequency = 90 rev/min
• Final frequency = 0 rev/sec
• Number of revolutions = 21

$\sf{\underline{\underline{\qquad \bigstar \: To \: Find \: \bigstar\qquad }}}$

• The time taken by the frictional torque is constant = ?

$\sf{\underline{\underline{\qquad \bigstar \: Solution \: \bigstar\qquad }}}$

$\tiny \bigstar \: \underline{\frak{First \: of \: all \: we \: have \: to \: convert \: initial \: frequency \: from \: rev/min \: to \: rev/sec :}}$

$:\implies \sf Initial \: frequency = \dfrac{90 \: rev}{1 \: min}$

$:\implies \sf Initial \: frequency = \dfrac{90 \: rev}{60 \: sec}$

$:\implies \sf Initial \: frequency = \dfrac{9 \: rev}{6 \: sec}$

$:\implies \underline{ \boxed{ \sf Initial \: frequency =1.5 \: rev/sec }}$

$\therefore\:\underline{\textsf{Initial frequency of rotating fan is \textbf{1.5 rev/sec}}}.$

$\tiny \bigstar \: \underline{\frak{Now, Let's \: calculate \: the \: time \: taken \: by \: the \: frictional \: torque :}}$

$\dashrightarrow\:\:\sf Number \: of \: revolutions =\Bigg\lgroup\dfrac{ Initial \: frequency - Final \: frequency}{2} \Bigg\rgroup\times Time$

$\dashrightarrow\:\:\sf 21 =\Bigg\lgroup\dfrac{1.5- 0}{2} \Bigg\rgroup\times Time$

$\dashrightarrow\:\:\sf 21 =\Bigg\lgroup\dfrac{1.5}{2} \Bigg\rgroup\times Time$

$\dashrightarrow\:\:\sf 21 \times 2 =1.5\times Time$

$\dashrightarrow\:\:\sf 42=1.5\times Time$

$\dashrightarrow\:\:\sf Time = \dfrac{42}{1.5}$

$\dashrightarrow\:\: \underline{ \boxed{\sf Time = 28 \: seconds}}$

$\therefore\:\underline{\textsf{The time taken by the frictional torque is \textbf{28 seconds}}}.$