# what is the si unit of Ks show yor answer in terms of base units using correct symbols.Show how worked it out

Es =1/2 ks X(squared)

Es = potential strain energy

Ks = spring constant

X = compression of spring measured in metres

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The usual (and clearer) way to write this relation is:

E = 1/2 k x^2

Note that E is energy (and therefore has units of energy), k is the spring constant, and x is a length, as you noted. Using dimensional analysis, we have the following:

Unit of energy in SI: Joules (J) = kg m^2/s^2

Unit of length in SI: meters.

[E] = [k] [x]^2

Where [A] means "dimension of A." [x] = m. [E] = kg m^2/s^2, etc.

kg m^2 /s^2 = [k] m^2

kg /s^2 = [k]

The units of the spring constant are kg /s^2 in the SI system. Since a Newton is a kg m/s^2, we also have that:

[k] = kg /s^2 = kg m/(m s^2) = 1/m (kg m)/s^2 = N/m

This is perhaps more intuitive; it suggests that a certain force is equivalent to a certain length. This is the force required to stretch the spring a particular length.

You can also see this from the Hooke spring law:

F = - kx

Briefly, the lesson here is that dimensions can be treated algebraically; two sides of an equation should have the same dimension. Therefore, I can treat them with the usual algebra rules - if I multiply one side by meters, I should multiply the other side by meters. This ensures the dimension of both sides stays the same.

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• just substitute the units in and solve for Ks....

Es = 1/2 ks x²

ks = 2 Es / x² = 2 x (potential energy ) / (m)²

but potential energy has units kg m²/s²

so ks = 2 x (kg m²/s²) / m² =2 kg / s²

ie units are kg/s²

or if you prefer....

since 1 N = 1 kg m / s²

ks = 2 x (kg m²/s²) / m² = 2 (kg m/s²) m / m² = 2 (kg m/s²) / m

= 2 N/m

ie units are N/m

so you pick. kg/s² or N/m both are correct.

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• The units need to be the same on both sides.

Ignore the 1/2, as it doesn't have units, it's just a number.

Rearrange the equation.

E/X^2 = k

The units of k are the same as the units of E/X^2.

The units of X are metres, so the units are whateer the units of E are over metres^2.

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• Ks would be in Newtons per meter. Es, the potential strain energy, is in Joules. One Joules is equal to one Newton times one meter. One Newton is equal to one kilogram times one meter per second per second. Since X is a distance, it is measured in meters. Because meters is squared, you need one meter in the denominator to cancel out the two in the numerator. Also, because energy is in Newton-meters, you need a Newton in the numerator. In this way, Ks is measured in Newtons per meter. In base units, a Newton is of course a kg times a meter per second per second. Therefore, if you divide kgm/s^2 by m, you get kg/s^2.

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