Units and Measurements (11Acd01)

Sheet 04 Finding the dimensions of Constant in Physical Relation.

1. The position of a particle moving along X-axis depends on time in accordance with the equation \(x=at^{2}+bt^{3}\), where x is in metre and t is in second. What are the units and dimensions of a and b?
   
   Ans: \(a=[LT^{-2}]\) and \(b=[LT^{-3}]\).
   
   
2. Write the dimensions of \(\frac{a}{b}\) in the relation \(F=a\sqrt{x}+bt^{2}\), where F is the force, x is the distance and t is time.
   
   Ans: \(\frac{a}{b}=\left[L^{\frac{1}{2}}T^{2}\right]\).
   
   
3. If, \(v=\frac{A}{t}+Bt^{2}+Ct^{3}\). Where; v is velocity, t is time and A,B,C are the constants then find dimensions of \(\frac{AB}{C}\)?
   
   Ans: \(\left[LT\right]\).
   
   
4. If, \(x=a+bt+ct^{2}\). Where; x is in metre and t is in seconds. What are the dimensions of a,b and c. Also write S.I units of a,b and c.
   
   Ans: \(a=[L]\), \(b=[LT^{-1}]\) and \(c=[LT^{-2}]\).
   
   
5. In Vander Wall’s equation \(\left[P+\frac{a}{V^{2}}\right]\left[V-b\right]=RT\). What are the dimensions and units of a and b?
   
   Ans: \([ML^{5}T^{-2}]\) and \([L^{3}]\).
   
   
6. What will be the dimension of \(a\times b\) in the relation \(P=\frac{b-x^{2}}{at}\). Where; P is power, x is distance and t is time.
   
   Ans: \(a\times b=[M^{-1}L^{2}T^{2}]\).
   
   
7. In the equation \(y=Asin\left(\omega t-kx\right)\). Where; x and t stands for distance and time respectively. Obtain dimensions of \(\omega\) and \(k\).
   
   Ans: \([\omega]=[M^{0}L^{0}T^{-1}]\) and \([a]=[ML^{5}T^{-2}]\).
   
   
8. If, x denotes the distance of an object and can be expressed as \(x=K\left[\frac{aCos\theta + bSin\theta}{a+b}\right]\). Then; what are the dimensions of K, if a and b has dimensions of lenght.
   
   Ans: \([L]\).