Dimensional Representation of Power using Fundamental Quantities
Understanding Dimensions
The goal is to express the dimensions of Power (P) in terms of Kinetic Energy (K), Current (I), and Charge (Q) as the fundamental quantities.
Deriving Base Dimensions
- Kinetic Energy (K): The dimension of energy is derived from $K = \frac{1}{2}mv^2$. Thus, the dimension is $[K] = [M][L]^2[T]^{-2}$.
- Current (I): This is taken as a fundamental quantity, so its dimension is $[I]$.
- Charge (Q): Charge is related to current and time by $Q = It$. Thus, the dimension is $[Q] = [I][T]$.
- Power (P): Power is the rate of change of energy, $P = \frac{Energy}{Time}$. Thus, the dimension is $[P] = \frac{[K]}{[T]}$.
Expressing Power in Fundamental Quantities
We need to express $[P] = \frac{[K]}{[T]}$ using K, I, and Q. From the dimension of charge, $[Q] = [I][T]$, we can find the dimension of time:
- $[T] = \frac{[Q]}{[I]} = [Q][I]^{-1}$.
Now, substitute the dimension of Time into the dimension of Power:
- $[P] = \frac{[K]}{[T]} = \frac{[K]}{[Q][I]^{-1}}$
- $[P] = [K][Q]^{-1}[I]$
- $[P] = [K I Q^{-1}]$
Therefore, the dimensional representation of power using K, I, and Q as fundamental quantities is $[K I Q^{-1}]$.
Similar Questions
- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$) - A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
A fly wheel of mass $10$ kg and radius $50$ cm is rotating at a rate of$360$ rpm. Assuming mass to be concentrated at the rim, the constant retarding torque required to stop the rotation of wheel in $6$ rotations will be x Nm. The value of x is ____________.
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Important Questions from Mechanics
- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$) - A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
- A body of density '$\rho$' is dropped slowly on the surface of a lake of depth $d$. If the density of the lake water be '$\rho'$' ($\rho' < \rho$) then the time taken by the body to reach the bottom of the lake is
Three blocks of masses $m_1 = 2\text{ kg}$, $m_2 = 3\text{ kg}$ and $m_3 = 5\text{ kg}$ are placed on a horizontal frictionless surface and a force of 30N pulls the system as shown below. The value of tension in the string between $m_2$ and $m_3$ will be
