Physics, asked by ayeshasalimshaikh097, 10 months ago

derive an expression for velocity at top and bottom position in vertical CM

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Answered by Anonymous

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derive an expression for velocity at top and bottom position in vertical CM

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ʟᴇᴛ ᴀ sᴍᴀʟʟ ʙᴏᴅʏ ᴏғ ᴍᴀss 'ᴍ' ᴀᴛᴛᴀᴄʜᴇᴅ ᴛᴏ ᴀ sᴛʀɪɴɢ ᴀɴᴅ ʀᴇᴠᴏʟᴠᴇᴅ ɪɴ ᴀ ᴠᴇʀᴛɪᴄᴀʟ ᴄɪʀᴄʟᴇ ᴏғ ʀᴀᴅɪᴜs 'ʀ'. ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ ᴡᴇɪɢʜᴛ ᴏғ ᴛʜᴇ ʙᴏᴅʏ ᴀʟᴡᴀʏs ᴀᴄᴛs ᴠᴇʀᴛɪᴄᴀʟʟʏ ᴅᴏᴡɴᴡᴀʀᴅ ᴀɴᴅ ᴛᴇɴsɪᴏɴ ɪɴ ᴛʜᴇ sᴛʀɪɴɢ ᴛᴏᴡᴀʀᴅs ᴛʜᴇ ᴄᴇɴᴛʀᴇ ᴏғ ᴛʜᴇ ᴄɪʀᴄᴜʟᴀʀ ᴘᴀᴛʜ.

ʟᴇᴛ ᴠ

ɪs ᴛʜᴇ sᴘᴇᴇᴅ ᴏғ ᴛʜᴇ ʙᴏᴅʏ ᴀɴᴅ ᴛ 2

ɪs ᴛʜᴇ ᴛᴇɴsɪᴏɴ ɪɴ ᴛʜᴇ sᴛʀɪɴɢ ᴀᴛ ᴛʜᴇ ʟᴏᴡᴇsᴛ ᴘᴏɪɴᴛ ʙ. sᴏ ᴀᴛ ᴛʜᴇ ʟᴏᴡᴇsᴛ ᴘᴏɪɴᴛ ᴛ 2 = ʀᴍᴠ 22 +ᴍɢ ...(1)

ᴛᴏᴛᴀʟ ᴇɴᴇʀɢʏ ᴀᴛ ʙᴏᴛᴛᴏᴍ=ᴘᴇ+ᴋᴇ=0+ 21 ᴍᴠ 2= 21

ᴍᴠ 22 ...(2)

ʟᴇᴛ ᴠ1

is ᴛʜᴇ sᴘᴇᴇᴅ ᴀɴᴅ ᴛ 1

ɪs ᴛʜᴇ ᴛᴇɴsɪᴏɴ ɪɴ ᴛʜᴇ sᴛʀɪɴɢ ᴀᴛ ʜɪɢʜᴇsᴛ ᴘᴏɪɴᴛ ᴀ.

sᴏ, ᴛ 1 = ʀᴍᴠ 12 −ᴍɢ ....(3)

ᴛᴏᴛᴀʟ ᴇɴᴇʀɢʏ ᴀᴛ ᴀ=ᴘᴇ+ᴋᴇ

=2ᴍɢʀ+ 21

ᴍᴠ 12 ....(4)

ғʀᴏᴍ ᴇϙᴜᴀᴛɪᴏɴs (1) ᴀɴᴅ (3)ᴛ 2 −ᴛ 1 = ʀᴍᴠ 22 +ᴍɢ−( ʀ

ᴍᴍᴠ1 −ᴍɢ) = ʀʀ (ᴠᴠ2 −ᴠ 12)+2ᴍɢ ...(5)

ʙʏ ʟᴀᴡ ᴏғ ᴄᴏɴsᴇʀᴠᴀᴛɪᴏɴ ᴏғ ᴇɴᴇʀɢʏ

ᴛᴏᴛᴀʟ ᴇɴᴇʀɢʏ ᴀᴛ ᴀ= ᴛᴏᴛᴀʟ ᴇɴᴇʀɢʏ ᴀᴛ ʙ.

ғʀᴏᴍ ᴇϙᴜᴀᴛɪᴏɴs (2) ᴀɴᴅ (4)

21 ᴍᴠ 22 = 21

ᴍᴠ 12 +2ᴍɢʀ

sᴏ ᴠ 22 −ᴠ 12 =4ʀɢ ...(6)

ᴘᴜᴛᴛɪɴɢ ᴛʜɪs ᴠᴀʟᴜᴇ ɪɴ ᴇϙᴜᴀᴛɪᴏɴ (5)

ᴛ 2 −ᴛ 1 = ʀᴍ

(4ʀɢ)+2ᴍɢ =4ᴍɢ+2ᴍɢ

ᴛ 2 −ᴛ 1

=6ᴍɢ

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