Question

then the maximum elastic strain energy in spring is ______.

Answer 1

A vertical spring mass system.

m = 0.2 kg. k = 40 N/m.

The forces acting on the mass are mg downwards and kx upwards.

m d^2 x/ dt^2 = mg - k x

The mass oscillates vertically about a mean position. It is a SHM. At the mean position the acceleration is 0 and speed is maximum.

So m g = k A
Amplitude A = mg/k.

At the lowest point in the oscillation, the PE is maximum and KE is 0. The mass m moves down a distance of 2 × A.

So maximum potential energy in the spring = mg × 2A using the energy conservation principle.

U = m g × 2 ( mg/k)
= 0.2 × 10× 2× 0.2 ×10/40 J
= 0.2 J

Answer 2

ᴀ ᴠᴇʀᴛɪᴄᴀʟ sᴘʀɪɴɢ ᴍᴀss sʏsᴛᴇᴍ.

ᴍ = 0.2 ᴋɢ. ᴋ = 40 ɴ/ᴍ.

ᴛʜᴇ ғᴏʀᴄᴇs ᴀᴄᴛɪɴɢ ᴏɴ ᴛʜᴇ ᴍᴀss ᴀʀᴇ ᴍɢ ᴅᴏᴡɴᴡᴀʀᴅs ᴀɴᴅ ᴋx ᴜᴘᴡᴀʀᴅs. 

ᴍ ᴅ^2 x/ ᴅᴛ^2 = ᴍɢ - ᴋ x

ᴛʜᴇ ᴍᴀss ᴏsᴄɪʟʟᴀᴛᴇs ᴠᴇʀᴛɪᴄᴀʟʟʏ ᴀʙᴏᴜᴛ ᴀ ᴍᴇᴀɴ ᴘᴏsɪᴛɪᴏɴ. ɪᴛ ɪs ᴀ sʜᴍ. ᴀᴛ ᴛʜᴇ ᴍᴇᴀɴ ᴘᴏsɪᴛɪᴏɴ ᴛʜᴇ ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ ɪs 0 ᴀɴᴅ sᴘᴇᴇᴅ ɪs ᴍᴀxɪᴍᴜᴍ. 

sᴏ ᴍ ɢ = ᴋ ᴀ
ᴀᴍᴘʟɪᴛᴜᴅᴇ ᴀ = ᴍɢ/ᴋ.

ᴀᴛ ᴛʜᴇ ʟᴏᴡᴇsᴛ ᴘᴏɪɴᴛ ɪɴ ᴛʜᴇ ᴏsᴄɪʟʟᴀᴛɪᴏɴ, ᴛʜᴇ ᴘᴇ ɪs ᴍᴀxɪᴍᴜᴍ ᴀɴᴅ ᴋᴇ ɪs 0. ᴛʜᴇ ᴍᴀss ᴍ ᴍᴏᴠᴇs ᴅᴏᴡɴ ᴀ ᴅɪsᴛᴀɴᴄᴇ ᴏғ 2 × ᴀ.

sᴏ ᴍᴀxɪᴍᴜᴍ ᴘᴏᴛᴇɴᴛɪᴀʟ ᴇɴᴇʀɢʏ ɪɴ ᴛʜᴇ sᴘʀɪɴɢ = ᴍɢ × 2ᴀ ᴜsɪɴɢ ᴛʜᴇ ᴇɴᴇʀɢʏ ᴄᴏɴsᴇʀᴠᴀᴛɪᴏɴ ᴘʀɪɴᴄɪᴘʟᴇ. 

ᴜ = ᴍ ɢ × 2 ( ᴍɢ/ᴋ)
= 0.2 × 10× 2× 0.2 ×10/40 ᴊ
= 0.2 ᴊ 


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