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assuming that mass M of the largest stone that can be moved by a flowing river depends 'v' velocity,densityof water and g acc. due to gravity.show that M varies with 6th power of velocity of flow....

11 years ago

M -> V, d (density) and g

Let M (proportional to) V^{a}d^{b}g^{c}

Now Hence M = k V^{a}d^{b}g^{c }where k is constant of proportionality

[M] = [V]^{a}[d]^{b}[g]^{c}

M^{1}L^{0}T^{0} = (L^{1}T^{-1})^{a}(M^{1}L^{-3})^{b}(L^{1}T^{-2})^{c}

M^{1}L^{0}T^{0} = M^{b }L^{a - 3b + c} T^{-a - 2c}

Comparing the powers,

b = 1

a - 3b + c = 0

Hence,

a - 3(1) + c = 0

a + c = 3 ...........1

-a - 2c = 0 ............2

Solving equations 1 and 2,

a + c = 3

-a - 2c = 0

---------------

-c = 3

Hence, c = - 3.

Substituting c = -3 in equation 1, we get a = 6.

Thus, M = k V^{6}dg

hence, it can be said that M varies with 6th power of velocity.

Hint: In any type of problems of dimensional analysis where it is given that This quantity depends on so and so... solve by this method always. It would be easier.

11 years ago

M is directly proportional to vˆx

M is directly proportional to dˆy

M is directly proportional to gˆz

m=k*v^x*d^y*g^z

m=(l*t^-1)^x*(m*l)^y*(lt^-2)^z

y=1

-x-2*z=0

z=-x/2

x-3*y-x/2=0

x=6

m=v^6

-3

2 years ago

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one year ago

M is directly proportional to vˆx

M is directly proportional to dˆy

M is directly proportional to gˆz

m=k*v^x*d^y*g^z

m=(l*t^-1)^x*(m*l)^y*(lt^-2)^z

y=1

-x-2*z=0

z=-x/2

x-3*y-x/2=0

x=6

m=v^6

-3

one year ago

Dear student,

Please find the attached answer to this question below.

Let M (proportional to) V^{a}d^{b}g^{c}

Now Hence M = k V^{a}d^{b}g^{c }where k is constant of proportionality

[M] = [V]^{a}[d]^{b}[g]^{c}

M^{1}L^{0}T^{0} = (L^{1}T^{-1})^{a}(M^{1}L^{-3})^{b}(L^{1}T^{-2})^{c}

M^{1}L^{0}T^{0} = M^{b }L^{a - 3b + c} T^{-a - 2c}

Comparing the powers,

b = 1

a - 3b + c = 0

Hence,

a - 3(1) + c = 0

a + c = 3 ...........1

-a - 2c = 0 ............2

Solving equations 1 and 2,

a + c = 3

-a - 2c = 0

---------------

-c = 3

Hence, c = - 3.

Substituting c = -3 in equation 1, we get a = 6.

Thus, M = k V^{6}dg

hence, it can be said that M varies with 6th power of velocity.

Hope it helps.

Thanks and regards,

Kushagra

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