The position of a particle moving under uniform acceleration is some function of time and the acceleration. Suppose we write this position as , where is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if and . Can this analysis give the value of ?
The expression for the position of the particle satisfies the values of , and the value cannot be determined.
The analysis of a relationship between different physical quantities by using the units of measurements and dimensions is called dimensional analysis. It is used to examine the correctness of an equation.
Dimensional analysis is used for verifying the correctness of anexpression and in changing the units from one system to another.
Write the expression for the position of the particle as a function of time and acceleration.
Here, is the position of the particle, is the acceleration, and are the integer values, is the constant, and is the time.
Write the individual dimensions for each physical quantity in the above equation.
The dimension of the position of the particle is .
Here, is the dimension of length.
The dimension of the acceleration of the particle is .
Here, T is the dimension of time.
The dimension of the time of the particle is T. Since is the constant, it has no dimensions.
Replace for , for , and T for in equation (1).
Replace for L in the left hand side of the above equation, which does not change the term L .
By equating the powers of the dimension L on both the sides of the above equation, we get:
Similarly, by equating the powers of the dimension T on both the sides of the equation, we get:
Since , replace the value of above to find the value.
From the above calculation of and the expression for the position of the particle is satisfied using the dimensions of the physical quantities.
Hence, the expression satisfies for the and values using the dimensional analysis, and since is a dimensionless constant, its value cannot be determined.
A crystalline solid consists of atoms stacked up in a repeating lattice structure. Consider a crystal as shown in Figure . The atoms reside at the corners of cubes of side . One piece of evidence for the regular arrangement of atoms comes from the flat surfaces along which a crystal separates, or cleaves, when it is broken. Suppose this crystal cleaves along a face diagonal as shown in Figure . Calculate the spacing d between two adjacent atomic planes that separate when the crystal cleaves.
94% of StudySmarter users get better grades.Sign up for free