Question

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Using dimensional method prove F=ma where F =force, m= mass and a= acceleration. ​

Answer 1

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✍️ꜰᴏʀᴄᴇ = ᴍᴀꜱꜱ × ᴀᴄᴄᴇʟᴇʀᴀᴛɪᴏɴ
✍️ꜰᴏʀᴄᴇ = ᴍᴀꜱꜱ × ᴠᴇʟᴏᴄɪᴛy/ᴛɪᴍᴇ

ᴏʀ, ᴡᴇ ᴄᴀɴ ᴡʀɪᴛᴇ,

✍️ᴍᴀꜱꜱ = ꜰᴏʀᴄᴇ×ᴛɪᴍᴇ / ᴠʀʟᴏᴄɪᴛy

ᴛʜᴇʀᴇꜰᴏʀᴇ, ᴡᴇ ɢᴇᴛ,

✍️[ᴍᴀꜱꜱ] = [ꜰᴛᴠ-¹]

Answer 2

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2nd derivation xD

Suppose an object of mass m has an initial velocity u. When a force of F is applied in the direction of its velocity for time t, its velocity becomes v.

∴The initial momentum of the object = mu,

Its final momentum after time t = mv.

∴Rate of change of momentum =

Change in momentum ÷ Time.

∴Rate of change of momentum =

(mv - mu) ÷ t = m (v-u) ÷ t = ma.

Acvording to Newton's second law of motion, the rate of change in momentum is proportional to the applied force.

∴ ma ∝ F

∴ F = k ma ( k = Constant of proportionality and its value is 1. )

F = m × a.