We have the equation U equals a t squared divided by the quantity b t over c plus t. Dimensional analysis is a fundamental tool in physics that ensures equations are mathematically consistent. We use basic dimensions: M for mass, L for length, and T for time. Our goal is to find the dimension of parameter b by applying dimensional consistency principles.
Let's analyze the equation U equals a t squared over b t over c plus t. We break this into components: the numerator contains a t squared, while the denominator has two terms being added together. A fundamental principle in mathematics is that only quantities with identical dimensions can be added or subtracted. This means b t over c must have the same dimension as t.
Now we apply dimensional consistency. Since terms being added must have identical dimensions, and t has dimension T, the term b t over c must also have dimension T. Writing this mathematically: the dimension of b t over c equals T. This gives us b times T over c equals T. Dividing both sides by T, we get b over c equals one, which means b has the same dimension as c.
Now we determine what c represents. Since we know b equals c dimensionally, we need to identify c's physical meaning. Looking at the equation structure, c appears as a characteristic time constant in the denominator. This is common in physics equations describing time-dependent processes. Therefore, c has dimension T, which means b also has dimension T. We can verify this: the denominator becomes T times T over T plus T, which equals T plus T, giving us T overall.
Let's perform a complete verification and explore examples. The equation structure appears in many physics contexts: RC circuits where b represents a time constant, exponential decay processes, and motion with resistance. Our dimensional analysis shows the numerator has dimension a times T squared, the denominator simplifies to T, giving an overall dimension of a times T. This confirms our result is correct. Therefore, the dimension of parameter b is T, representing time.