we have learnt that Earth's gravitational field strength is 10 N/kg is the same as its acceleration due to free fall (10 m/s/s. even though their units are different, they are said to be dimentionally the same. prove that N/kg is the same as m/s/s
If you have learned it is 10N/kg, you have learned it wrong. Nowhere on Earth is it that value.
Force=mass*acceleration
Newtons=kg*g
g= N/kg, this is the same g you know as 9.8m/s^2
To prove that Newtons per kilogram (N/kg) is the same as meters per second squared (m/s²), we can start by recalling the definitions of these quantities.
The gravitational field strength (g) of Earth is given as 10 N/kg. This means that for every kilogram of mass, there is a force of 10 Newtons acting on it due to the Earth's gravitational pull.
On the other hand, acceleration due to free fall (a) is given as 10 m/s². This represents the change in velocity per second when an object falls freely under the influence of gravity.
Now, let's examine the units of N/kg and m/s²:
- N (Newton) is the unit of force
- kg (kilogram) is the unit of mass
- m (meter) is the unit of distance
- s (second) is the unit of time
Notice that in Newtons per kilogram (N/kg), the unit of force (N) is divided by the unit of mass (kg). This division effectively cancels out the unit of mass, leaving only the unit of force (N).
Similarly, in meters per second squared (m/s²), the unit of distance (m) is divided by the square of the unit of time (s). In this case, the division cancels out the unit of time squared, leaving only the unit of distance.
So, even though the units N/kg and m/s² are different, they represent the same physical quantity, which is the gravitational acceleration experienced by objects near the Earth's surface.
To prove that Newton per kilogram (N/kg) is the same as meters per second squared (m/s²), we need to demonstrate that their dimensions are equivalent.
Dimensions are the physical quantities used to describe the nature of a particular quantity. In this case, we have two quantities: force and acceleration.
The dimension of force is [M.L.T⁻²] (mass × length × time⁻²), and it is represented by the unit Newton (N). This unit originates from Newton's second law of motion: F = ma, where F represents force, m represents mass, and a represents acceleration.
The dimension of acceleration is [L.T⁻²] (length × time⁻²), and it is represented by the unit meters per second squared (m/s²). Acceleration describes the rate of change of velocity over time.
Now, let's compare the dimensions of N/kg and m/s²:
1. N/kg:
- N represents the dimension [M.L.T⁻²] (force).
- kg represents the dimension [M] (mass).
Therefore, the overall dimension of N/kg is [M.L.T⁻²] / [M], which simplifies to L.T⁻².
Consequently, N/kg has the dimension of m/s² (L.T⁻²), which is the same as acceleration.
Hence, we have proven that N/kg is indeed equivalent to m/s², despite the difference in their units.