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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....

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....

Grade:

5 Answers

Vishrant Vasavada
23 Points
11 years ago

M -> V, d (density) and g

 

    Let M (proportional to) Vadbgc

Now Hence M = k Vadbgwhere k is constant of proportionality

[M] = [V]a[d]b[g]c

M1L0T0 = (L1T-1)a(M1L-3)b(L1T-2)c

M1L0T0 = Mb La - 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 V6dg

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.

 

dip ak
16 Points
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

Saksham agarwal
15 Points
2 years ago
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mahi
13 Points
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

Kushagra Madhukar
askIITians Faculty 629 Points
one year ago
Dear student,
Please find the attached answer to this question below.
 

Let M (proportional to) Vadbgc

Now Hence M = k Vadbgwhere k is constant of proportionality

[M] = [V]a[d]b[g]c

M1L0T0 = (L1T-1)a(M1L-3)b(L1T-2)c

M1L0T0 = Mb La - 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 V6dg

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

 

Hope it helps.

Thanks and regards,

Kushagra

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