1.9 If P is a force and x a length, what are the dimensions (in the F L T system ) of (a) d P /

1.9 If P is a force and x a length, what are the dimensions...

Key Concepts

Integration in Physical Contexts

Integration is the inverse process of differentiation, representing the accumulation of a physical quantity over an interval of another variable, typically resulting in the multiplication of the original units with those of the integrating variable. In this context, integrating a force with respect to a length effectively appends a length dimension to the force unit, resulting in a quantity with dimensions that are the product of force and length.

Dimensional Analysis

Dimensional analysis serves as a tool to verify the consistency of physical equations by expressing quantities in terms of their fundamental units, in this case using the F (force), L (length), and T (time) system. It ensures that both sides of an equation have the same dimensions, thereby helping in deriving relationships between physical quantities and checking the validity of computations.

Differentiation in Physical Contexts

Differentiation involves finding the rate of change of a physical quantity with respect to another quantity. When differentiating a force with respect to a length, the operation leads to the division of the force dimension by the length, modifying the original units appropriately. Successive differentiations with respect to length further increase the power of L in the denominator, reflecting higher order rate changes.

Transcript

00:01 So in this question, what we are required to do is that we are required to find out the dimensions, right? and this time we need to find out the dimensions in the flt system, right? so there are three cases.

00:20 The first case is the derivative of p with respect to x, right? in the second case, we have triple derivative of p with respect to x.

00:31 Right and then we have the integral of p dx right so let's start with the solution so the first step in order to solve this question is to analyze the dimensions of the quantities that are given to us right that is force and length right so let's take a look at the dimensional analysis that we have for this question right now let me make a table right not a complete table but a table so that way it would be easy for you to understand right so this is the first row and this first row has two columns so in the first column we have the physical quantities right since i have to not have i do not have a space over here so i'm just going to write q for quantity right and in the second column we have the dimensions for the philpd system right so the first quantity is force right and force has a unit of newton right so the dimension for force it's f right simple it's force because as fl t itself it's force length time right so the dimension is force.

02:15 Now, moving forward, we have the second quantity that is length, right? it's very simple.

02:25 The unit of length is meter and its dimension is length, right? so let's move forward towards the cases that we have.

02:35 Now, let's take a look at the first case in which there is a derivative involved, right? so derivative of p with respect to x, right? right so what you need to do is i just plug in the um dimensions that we just discussed so the dimension of p is f and the dimension of x is l right so as you can see over here there is division involved right so when we move this l over here so this would be equal to f per length or f times negative f times l raised up negative one right so this is the dimension for the first part right it was it was very easy we just plugged in the dimensions right now one thing to note over here is that you don't need to actually take the derivative right you just need to plug in the dimensions right the dimensions or dimensions of the physics of the physics of the physics quantities that are involved right so this is the answer for the first part of first case right then the b part is that triple derivative of b with respect to q x right so again you do not need to take the derivative you just need to plug in the dimension right and it would be the same as part so that is f divided by length cube right so now notice one thing that oh in the numerator right in the numerator the cube was with the derivative and not with the physical quantity right and in the numerator this cube is with the physical quantity right so therefore we have a cube with the dimension as bit right so again you need to shift this in over here so when you do that you'll get that it's force per length cube or force times l ways to 1 negative 3 so this is a solution for part b now moving forward we have our last part that is the integral of b times d x right so again right we have two physical quantities.

05:19 The first is p and the second is x.

05:22 Right.

05:23 So what you need to do is that just write down the physical quantity...

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