Question

If the unit of force, energy and velocity are 20N, 200 J and 5 m/s. find r=the unit of mass length and time.

Answer 1

Solution:

Given:

=> Force = 20 N

=> Energy = 200 J

=> Velocity = 5 m/s

To Find:

=> Unit of mass, length and time.

Formula used:

$\sf{\implies [M^{1}L^{1}T^{-2}]\;\;\;\;(Dimensional\;formula\;of\;force)}$

$\sf{\implies [M^{1}L^{2}T^{-2}]\;\;\;\;(Dimensional\;formula\;of\;Energy)}$

$\sf{\implies [M^{0}L^{1}T^{-1}]\;\;\;\;(Dimensional\;formula\;of\;Velocity)}$

So,

$\sf{\implies M^{1}L^{1}T^{-2}=20\;\;\;\;.........(1)}$

$\sf{\implies M^{1}L^{2}T^{-2}=200\;\;\;\;.........(2)}$

$\sf{\implies M^{0}L^{1}T^{-1}=5\;\;\;\;.........(3)}$

Now, compair Equation (1) and (2), we get

$\sf{\implies \dfrac{M^{1}L^{2}T^{-2}}{M^{1}L^{1}T^{-2}}=\dfrac{200}{20}}$

$\large{\boxed{\boxed{\blue{\sf{\implies L = 10}}}}}$

Put the value of L in Equation (3), we get

$\sf{\implies M^{0}L^{1}T^{-1}=5}$

$\sf{\implies 10\times \dfrac{1}{T}=5}$

$\sf{\implies T = \dfrac{10}{5}}$

$\large{\boxed{\boxed{\red{\sf{\implies T = 2}}}}}$

Put the value of L and T in Equation (1), we get

$\sf{\implies M^{1}L^{1}T^{-2}=20}$

$\sf{\implies M\times 10\times \bigg(\dfrac{1}{2}\bigg)^{2}=20}$

$\sf{\implies M\times 10\times \dfrac{1}{4}=20}$

$\sf{\implies M \times \dfrac{10}{4} =20}$

$\large{\boxed{\boxed{\green{\sf{\implies M = 8}}}}}$

So, unit of

=> Mass = 8 kg

=> Length = 10 m

=> Time = 2 sec

Answer 2

${\bold{\underline{\underline{Answer:}}}}$

${\bold{\therefore Mass=8\:Kg}}$

${\bold{\therefore Length=10\:m}}$

${\bold{\therefore Time=2\:sec}}$

${\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

• In the given question information given about the unit of force, energy and velocity are 20N, 200 J and 5 m/s.

• We have to find the unit of mass length and time.

$\underline \bold{Given : } \\ \implies Force= 20 \: N \\ \\ \implies Energy = 200 \: J\\ \\ \implies Velocity = 5 \: m/s \\ \\ \underline \bold{To \: Find : } \\ \implies Unit \: of \: Mass = ? \\ \\ \implies Unit \:o f \: Length = ? \\ \\ \implies Unit \: of \: Time = ?$

• According to given question :

$\bold{By \: usin g\: Dimensions : } \\ \implies Force = [{M}^{1} {L}^{1} {T}^{ - 2}] \\ \\ \implies 20 =[ {M}^{1} {L}^{1} {T}^{ - 2} ]- - - - - (1) \\ \\ \bold{For \: Energy : } \\ \implies Energy = [{M}^{1} {L}^{2} {T}^{-2} ]</p><p>\\\\ \implies 200=[{M}^{1} {L}^{2} {T}^{-2}] - - - - - (2)\\ \\ \bold{For \: Velocity : } \\ \implies Velocity = [{M}^{0} {L}^{1} {T}^{-1}] \\ \\ \implies 5 = [{M}^{0} {L}^{1} {T}^{-1} ]- - - - - (3) \\ \\ \bold{dividing\: (1) \: and \: (2)} \\ \\ \implies \frac{ \cancel{{M}^{1}} {L}^{2} \cancel{{T}^{-2} } } { \cancel{{M}^{1} } {L}^{1} \cancel{{T}^{ - 2}}} = \frac{ \cancel{200}}{ \cancel{20}} \\ \\ \implies {L}^{2 - 1} = 10 \\ \\ \bold{\implies {L} = 10 \: m} \\ \\ \bold{Putting \: value \: f \: l \: in \: (2)} \\ \implies \: 10[{M}^{0} {T}^{-1} ]= 5 \\ \\ \implies {T}^{ - 1} = \frac{ \cancel5}{ \cancel{10}} \\ \\ \bold{\implies T = 2 \: sec } \\ \\ \bold{Putting \: value \: of \:T \:a nd \: L\: in \:(1)} \\ \implies {M}^{1} {L}^{1} {T}^{ - 2}= 20 \\ \\ \implies [{M}^{1}]10 \times (\frac{1}{2} )^{2} = 20 \\ \\ \implies[{M}^{1}] = 2\times 4 \\ \\ \bold{ \implies {M}^{1} = 8 \: kg}$