Question

A cylindrical piece of cork of density of base area A and height h floats in a liquid of density .ρ₁ The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period T=2π√hρ/ρ₁g where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).

Answer 1

ʙᴀsᴇ ᴀʀᴇᴀ ᴏғ ᴛʜᴇ ᴄᴏʀᴋ = ᴀ

ʜᴇɪɢʜᴛ ᴏғ ᴛʜᴇ ᴄᴏʀᴋ = ʜ

ᴅᴇɴsɪᴛʏ ᴏғ ᴛʜᴇ ʟɪǫᴜɪᴅ = ρ1
$t = 2\pi \sqrt{ \frac{ahp}{ap1g} } = 2\pi \sqrt{ \frac{hp}{p1g} }$


ᴅᴇɴsɪᴛʏ ᴏғ ᴛʜᴇ ᴄᴏʀᴋ = ρ

ɪɴ ᴇǫᴜɪʟɪʙʀɪᴜᴍ:
ᴡᴇɪɢʜᴛ ᴏғ ᴛʜᴇ ᴄᴏʀᴋ = ᴡᴇɪɢʜᴛ ᴏғ ᴛʜᴇ ʟɪǫᴜɪᴅ ᴅɪsᴘʟᴀᴄᴇᴅ ʙʏ ᴛʜᴇ ғʟᴏᴀᴛɪɴɢ ᴄᴏʀᴋ
ʟᴇᴛ ᴛʜᴇ ᴄᴏʀᴋ ʙᴇ ᴅᴇᴘʀᴇssᴇᴅ sʟɪɢʜᴛʟʏ ʙʏ x. ᴀs ᴀ ʀᴇsᴜʟᴛ, sᴏᴍᴇ ᴇxᴛʀᴀ ᴡᴀᴛᴇʀ ᴏғ ᴀ ᴄᴇʀᴛᴀɪɴ ᴠᴏʟᴜᴍᴇ ɪs ᴅɪsᴘʟᴀᴄᴇᴅ. ʜᴇɴᴄᴇ, ᴀɴ ᴇxᴛʀᴀ ᴜᴘ-ᴛʜʀᴜsᴛ ᴀᴄᴛs ᴜᴘᴡᴀʀᴅ ᴀɴᴅ ᴘʀᴏᴠɪᴅᴇs ᴛʜᴇ ʀᴇsᴛᴏʀɪɴɢ ғᴏʀᴄᴇ ᴛᴏ ᴛʜᴇ ᴄᴏʀᴋ.
ᴜᴘ-ᴛʜʀᴜsᴛ = ʀᴇsᴛᴏʀɪɴɢ ғᴏʀᴄᴇ, ғ = ᴡᴇɪɢʜᴛ ᴏғ ᴛʜᴇ ᴇxᴛʀᴀ ᴡᴀᴛᴇʀ ᴅɪsᴘʟᴀᴄᴇᴅ
ғ = ­–(ᴠᴏʟᴜᴍᴇ × ᴅᴇɴsɪᴛʏ × ɢ)
ᴠᴏʟᴜᴍᴇ = ᴀʀᴇᴀ × ᴅɪsᴛᴀɴᴄᴇ ᴛʜʀᴏᴜɢʜ ᴡʜɪᴄʜ ᴛʜᴇ ᴄᴏʀᴋ ɪs ᴅᴇᴘʀᴇssᴇᴅ
ᴠᴏʟᴜᴍᴇ = ᴀx
∴ ғ = – ᴀ x ρ1 ɢ …..(ɪ)
ᴀᴄᴄʀᴏᴅɪɴɢ ᴛᴏ ᴛʜᴇ ғᴏʀᴄᴇ ʟᴀᴡ:
ғ = ᴋx 
ᴋ = ғ/x
ᴡʜᴇʀᴇ, ᴋ ɪs ᴄᴏɴsᴛᴀɴᴛ
ᴋ = ғ/x = -ᴀρ1 ɢ ….(ɪɪ)
ᴛʜᴇ ᴛɪᴍᴇ ᴘᴇʀɪᴏᴅ ᴏғ ᴛʜᴇ ᴏsᴄɪʟʟᴀᴛɪᴏɴs ᴏғ ᴛʜᴇ ᴄᴏʀᴋ:
ᴛ = 2π √ᴍ/ᴋ      ….(ɪɪɪ)
ᴡʜᴇʀᴇ,
ᴍ = ᴍᴀss ᴏғ ᴛʜᴇ ᴄᴏʀᴋ
= ᴠᴏʟᴜᴍᴇ ᴏғ ᴛʜᴇ ᴄᴏʀᴋ × ᴅᴇɴsɪᴛʏ
= ʙᴀsᴇ ᴀʀᴇᴀ ᴏғ ᴛʜᴇ ᴄᴏʀᴋ × ʜᴇɪɢʜᴛ ᴏғ ᴛʜᴇ ᴄᴏʀᴋ × ᴅᴇɴsɪᴛʏ ᴏғ ᴛʜᴇ ᴄᴏʀᴋ
= ᴀʜρ
ʜᴇɴᴄᴇ, ᴛʜᴇ ᴇxᴘʀᴇssɪᴏɴ ғᴏʀ ᴛʜᴇ ᴛɪᴍᴇ ᴘᴇʀɪᴏᴅ ʙᴇᴄᴏᴍᴇs: