1.

If momentum, time and energy were chosen as basic quantities, find dimensions of (a) mass and (b) force.

Answer» The dimension for momentum, time and energy are
`p = [MLT^(-1)]`
`t = [T]`
`E = [ML^(2) T^(-2)]`
(a) `m prop p^(a) t^(b) E^(c )`
`M prop [MLT^(-1)]^(a) [T]^(b) [ML^(2) T^(-2)]^(c )`
`M^(1) L^(0) T^(0) prop M^(a + c) L^(a + 2c) T^(-a + b - 2c)`
Comparing power of `M, L` and `T`
`a + c = 1`
`a + 2c = 0`
`-a + b - 2c = 0`
Solving we get
`C = - 1, a = 2, b = 0`
`m prop p^(2) E^(-1)`
(b) `F prop p^(a) t^(b) E^(c )`
`MLT^(-2) prop [MLT^(-1)]^(a) [T]^(b) [ML^(2) T^(-2)]^(c)`
`M^(1) L^(1) T^(-2) prop M^(a + c) L^(a + 2c) T^(- a + b + 2c)`
Comparing power of `M, L` and `T`
`a + c = 1`
`a + 2c = 1`
`-a + b - 2c = -2`
`c = 0, a = 1, b = - 1`
`F prop p t^(-1)`


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