Find the dimensions of the following:
Moment of a couple
Power
Momentum
Velocity and
Density
The dimensions of the following quantities are as follows:
- Moment of a couple: The moment of a couple has dimensions of force multiplied by length. Therefore, its dimensions are [M L^2 T^-2].
- Power: Power is the rate at which work is done or energy is transferred. It is defined as the work done or energy transferred divided by the time taken. The dimensions of power are given by the dimensions of work or energy divided by the dimensions of time. Therefore, its dimensions are [M L^2 T^-3].
- Momentum: Momentum is the product of mass and velocity. The dimensions of momentum are given by the dimensions of mass multiplied by the dimensions of velocity. Therefore, its dimensions are [M L T^-1].
- Velocity: Velocity is the rate of change of displacement or the rate at which an object moves in a specific direction. The dimensions of velocity are given by the dimensions of displacement divided by the dimensions of time. Therefore, its dimensions are [L T^-1].
- Density: Density is defined as the mass per unit volume of a substance. The dimensions of density are given by the dimensions of mass divided by the dimensions of volume. Therefore, its dimensions are [M L^-3].
To find the dimensions of each quantity, we need to express them in terms of their fundamental dimensions. Let's break it down step by step:
1. Moment of a couple:
The moment of a couple is a measure of the turning effect created by two equal and opposite forces acting on a body at different points. The dimensions of the moment of a couple are given by the product of the force and the perpendicular distance between the points of application of the forces. Therefore, the dimensions of the moment of a couple are [Force] × [Length].
2. Power:
Power is the rate at which work is done or energy is transferred. It is given by the formula Power = Work/Time. The dimensions of power can be derived by substituting the dimensions of work and time into the formula. Work is given by the product of force and displacement, so its dimensions are [Force] × [Length]. Time has dimensions of [Time]. Therefore, the dimensions of power are [Force] × [Length] / [Time].
3. Momentum:
Momentum is a measure of an object's motion and is defined as the product of its mass and velocity. The dimensions of momentum are given by the product of mass and velocity. Mass has dimensions [Mass] and velocity has dimensions [Length] / [Time]. Therefore, the dimensions of momentum are [Mass] × [Length] / [Time].
4. Velocity:
Velocity refers to the rate of change of displacement with respect to time. The dimensions of velocity are given by the ratio of displacement ([Length]) to the time taken ([Time]). Therefore, the dimensions of velocity are [Length] / [Time].
5. Density:
Density is the mass per unit volume of a substance. It is given by the formula Density = Mass/Volume. The dimensions of density can be derived by substituting the dimensions of mass and volume into the formula. Mass has dimensions [Mass], and volume has dimensions [Length]^3. Therefore, the dimensions of density are [Mass] / [Length]^3.
So, the dimensions of the given quantities are:
- Moment of a couple: [Force] × [Length]
- Power: [Force] × [Length] / [Time]
- Momentum: [Mass] × [Length] / [Time]
- Velocity: [Length] / [Time]
- Density: [Mass] / [Length]^3