Question

derive an expression for the lower speed limit and upper speed limit for the vehicle moving along the banked road​

Answer 1

Answer:

Explanation:

ʙᴀɴᴋɪɴɢ ᴏғ ʀᴏᴀᴅ : ᴛᴏ ᴀᴠᴏɪᴅ ʀɪᴋ ᴏғ ᴋɪᴅᴅɪɴɢ ᴀ ᴡᴇʟʟ ᴀ ᴛᴏ ʀᴇᴅᴜᴄᴇ ᴛʜᴇ ᴡᴇᴀʀ ᴀɴᴅ ᴛᴇᴀʀ ᴏғ ᴛʜᴇ ᴄᴀʀ ᴛʏʀᴇ, ᴛʜᴇ ʀᴏᴀᴅ ᴜʀғᴀᴄᴇ ᴀᴛ ᴀ ʙᴇɴᴅ ɪ ᴛɪʟᴛᴇᴅ ɪɴᴡᴀʀᴅ, ɪ.ᴇ., ᴛʜᴇ ᴏᴜᴛᴇʀ ɪᴅᴇ ᴏғ ʀᴏᴀᴅ ɪ ʀᴀɪᴇᴅ ᴀʙᴏᴠᴇ ɪᴛ ɪɴɴᴇʀ ɪᴅᴇ. ᴛʜɪ ɪ ᴄᴀʟʟᴇᴅ 'ʙᴀɴᴋɪɴɢ ᴏғ ʀᴏᴀᴅ'. ᴄᴏɴɪᴅᴇʀ ᴀ ᴄᴀʀ ᴛᴀᴋɪɴɢ ᴀ ʟᴇғᴛ ᴛᴜʀɴ ᴀʟᴏɴɢ ᴀ ʀᴏᴀᴅ ᴏғ ʀᴀᴅɪᴜ ʀ ʙᴀɴᴋᴇᴅ ᴀᴛ ᴀɴ ᴀɴɢʟᴇ θ ғᴏʀ ᴀ ᴅᴇɪɢɴᴇᴅ ᴏᴘᴛɪᴍᴜᴍ ᴘᴇᴇᴅ ᴠ. ʟᴇᴛ ᴍ ʙᴇ ᴛʜᴇ ᴍᴀ ᴏғ ᴛʜᴇ ᴄᴀʀ. ɪɴ ɢᴇɴᴇʀᴀʟ, ᴛʜᴇ ғᴏʀᴄᴇ ᴀᴄᴛɪɴɢ ᴏɴ ᴛʜᴇ ᴄᴀʀ ᴀʀᴇ: (ᴀ) ɪᴛ ᴡᴇɪɢʜᴛ   ᴍɢ ​   , ᴀᴄᴛɪɴɢ ᴠᴇʀᴛɪᴄᴀʟʟʏ ᴅᴏᴡɴ (ʙ) ᴛʜᴇ ɴᴏʀᴍᴀʟ ʀᴇᴀᴄᴛɪᴏɴ ᴏғ ᴛʜᴇ ʀᴏᴀᴅ   ɴ  , ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ᴛᴏ ᴛʜᴇ ʀᴏᴀᴅ ᴜʀғᴀᴄᴇ (ᴄ) ᴛʜᴇ ғʀɪᴄᴛɪᴏɴᴀʟ ғᴏʀᴄᴇғ  s ​    ​    ᴀʟᴏɴɢ ᴛʜᴇ ɪɴᴄʟɪɴᴇᴅ sᴜʀғᴀᴄᴇ ᴏғ ᴛʜᴇ ʀᴏᴀᴅ. ʀᴇsᴏʟᴠᴇ   ɴ   ᴀɴᴅ   ғ  s ​    ​    ɪɴᴛᴏ ᴛᴡᴏ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ᴄᴏᴍᴘᴏɴᴇɴᴛs ɴᴄᴏsθ ᴠᴇʀᴛɪᴄᴀʟʟʏ ᴜᴘ ᴀɴᴅ   ғ  s ​    ​   sɪɴθ ᴠᴇʀᴛɪᴄᴀʟʟʏ ᴅᴏᴡɴ, ɴsɪɴθ ᴀɴᴅ   ғ  s ​    ​   ᴄᴏsθ ʜᴏʀɪᴢᴏɴᴛᴀʟʟʏ ᴛᴏᴡᴀʀᴅs ᴛʜᴇ ᴄᴇɴᴛʀᴇ ᴏғ ᴛʜᴇ ᴄɪʀᴄᴜʟᴀʀ ᴘᴀᴛʜ. ɪғ ᴠ  ᴍᴀx ​    ɪs ᴛʜᴇ ᴍᴀxɪᴍᴜᴍ sᴀғᴇ sᴘᴇᴇᴅ ᴡɪᴛʜᴏᴜᴛ sᴋɪᴅᴅɪɴɢ.  ʀ ᴍᴠ  ᴍᴀx  ​    ​   =ɴsɪɴθ+ғ  s ​   ᴄᴏsθ                 =ɴsɪɴθ+μ  s ​   ɴᴄᴏsθ  ʀ ᴍᴠ  ᴍᴀx  ​    ​   =ɴ(sɪɴθ+μ  s ​   ᴄᴏsθ)....()  ᴀɴᴅ  ɴᴄᴏsθ=ᴍɢ+ғ  s ​   sɪɴθ                =ᴍɢ+μ  s ​   ɴsɪɴθ    ∴ᴍɢ=ɴ(ᴄᴏsθ−μ  s ​   sɪɴθ)...()  ᴅɪᴠɪᴅɪɴɢ ᴇǫ. () ʙʏ ᴇǫ. (), ʀ.ᴍɢ ᴍᴠ  ᴍᴀx  ​    ​   =  ɴ(ᴄᴏsθ−μ  s ​   sɪɴθ) ɴ(sɪɴθ+μ  s ​   ᴄᴏsθ) ​     ∴  ʀɢ ᴠ  ᴍᴀx  ​    ​   =  ᴄᴏsθ−μ  s ​   sɪɴθ sɪɴθ+μ  s ​   ᴄᴏsθ ​   =  −μ  s ​   ᴛᴀɴθ ᴛᴀɴθ+μ  s ​    ​     ∴ᴠ  ᴍᴀx ​   =  −μ  s ​   ᴛᴀɴθ ʀɢ(ᴛᴀɴθ+μ  s ​   ) ​    ​   ...,.()  ᴛʜɪs ɪs ᴛʜᴇ ᴇxᴘʀᴇssɪᴏɴ ғᴏʀ ᴛʜᴇ ᴍᴀxɪᴍᴜᴍ sᴀғᴇ sᴘᴇᴇᴅ ᴏɴ ᴀ ʙᴀɴᴋᴇᴅ ʀᴏᴀᴅ. ᴀᴛ ᴛʜᴇ ᴏᴘᴛɪᴍᴜᴍ sᴘᴇᴇᴅ, ᴛʜᴇ ғʀɪᴄᴛɪᴏɴ ʙᴇᴛᴡᴇᴇɴ ᴛʜᴇ ᴄᴀʀ ᴛʏʀᴇs ᴀɴᴅ ᴛʜᴇ ʀᴏᴀᴅ sᴜʀғᴀᴄᴇ ɪs ɴᴏᴛ ᴄᴀʟʟᴇᴅ ɪɴᴛᴏ ᴘʟᴀʏ. ʜᴇɴᴄᴇ, ʙʏ sᴇᴛᴛɪɴɢ μ  s ​   = ɪɴ ᴇǫ. (), ᴛʜᴇ ᴏᴘᴛɪᴍᴜᴍ sᴘᴇᴇᴅ ᴏɴ ᴀ ʙᴀɴᴋᴇᴅ ᴄɪʀᴄᴜʟᴀʀ ʀᴏᴀᴅ ɪs ᴠ=  ʀɢᴛᴀɴθ ​   ...() ∴ᴛᴀɴθ=  ʀɢ ᴠ     ​    ᴏʀ θ=ᴛᴀɴ  −  (  ʀɢ ᴠ     ​   ) ғʀᴏᴍ ᴛʜɪs ᴇǫ. ᴡᴇ sᴇᴇ ᴛʜᴀᴛ θ ᴅᴇᴘᴇɴᴅs ᴜᴘᴏɴ ᴠ,ʀ ᴀɴᴅ ɢ. ᴛʜᴇ ᴀɴɢʟᴇ ᴏғ ʙᴀɴᴋɪɴɢ ɪs ɪɴᴅᴇᴘᴇɴᴅᴇɴᴛ ᴏғ ᴛʜᴇ ᴍᴀss ᴏғ ᴀ ᴠᴇʜɪᴄʟᴇ ɴᴇɢᴏᴛɪᴀᴛɪɴɢ ᴛʜᴇ ᴄᴜʀᴠᴇ.