1) use dimensional analysis to determine which of the following expressions gives the area of a circle: is it pie r ^2, or 2 pie r? explain
- pie r^2 right?? because its what they normally use to find area of a circle
2) If a distance d has units of meters and a time T has units of sec, does the quantity T+d make sense physically?? what about the quantity d/T? explain in both cases
- ?? I don't know
1) To determine which of the two expressions gives the area of a circle, we can use dimensional analysis. In dimensional analysis, we analyze the units of measurement to check if they are consistent.
First, let's consider the expression "pie r^2." Here, "r" represents the radius of the circle, and "pie" represents the mathematical constant pi. The units of "r" are in meters (m), and the square of "r^2" will be in square meters (m^2).
Now, let's consider the expression "2 pie r." Here, "2 pie" is just a scalar constant, and "r" represents the radius of the circle. Similarly, the units of "r" are in meters (m), and the units of "2 pie r" will also be in meters (m).
Thus, based on dimensional analysis, both expressions have units of meters (m), which is consistent with measuring the area of a circle. However, the correct expression for the area of a circle is "pie r^2." This is the formula commonly used and derived from mathematical principles. So, the correct answer is "pie r^2" because it represents the actual mathematical formula for calculating the area of a circle.
2) The quantity "T + d" represents adding a distance "d" in meters (m) to a time "T" in seconds (s). Combining distance and time in this way does not make physical sense because we are adding two quantities with different units. Distances and times are fundamentally different types of measurements and cannot be added together in a physically meaningful way.
On the other hand, the quantity "d/T" represents dividing a distance "d" in meters (m) by a time "T" in seconds (s). This ratio represents the speed or velocity of an object, which is a physically meaningful quantity. Dividing a distance by time gives us a rate of change in distance over time, which is commonly used in physics.
So, "T + d" does not make physical sense, while "d/T" represents a physically meaningful quantity related to speed or velocity.