Final answer:
The dimensions of (a/b) in the equation P = (a - ct²)/bx are MLT⁻², given that P is pressure with dimensions ML⁻¹T⁻², c has dimensions of ML⁻¹T⁻², t is time (T), and x is displacement (L).
Step-by-step explanation:
To determine the dimensions of (a/b) given P = (a - ct²)/bx where P is pressure, we need to express everything in terms of their fundamental dimensions. Recalling that pressure (P) has dimensions of ML⁻¹T⁻², we can do a dimensional analysis assuming each term in the equation has the same dimensions as P.
We rewrite the given equation as P = a/bx - ct²/bx. Now, x has dimensions of L (length), and time t has dimensions of T (time). Since ct² has the same pressure dimension, c must have dimensions of ML⁻¹T⁻². Multiplying these by T² (from t²), we find that the dimensions of c are indeed M·L⁻¹.
So, simplifying the original equation, we find that a must have dimensions of ML⁻¹T⁻²·L = MLT⁻². Dividing by b, which is dimensionless since it is the ratio of two like dimensions, we find that the dimensions of (a/b) are MLT⁻². This also explains that a is related to force and b is a dimensionless coefficient.