An object tied to the end of a string moves in a circle. The force exerted by the string depends on mass of the object, it's speed, and the radius of a circle. What combination of these variables gives the dimensions ( ML/T^2) for the force?
F varies directly as mvr
F=Km^av^br^c
MLT^-2= KM^a(LT^-1)^bL^c
M=M^a
a=1 ......(1)
L=L^b+c
b+c=1 ....(2)
T^-2= T^-b
b=2 .......(3)
From (2)
b+c=1
2+c=1
c=-1 .......(4)
Subst. values of a,b and c in the eqn
F=KMV^2r^-1
F=KMV^2/r
F varies as MV^2/r
Please respond to que.
To determine the combination of variables that gives the dimensions of force (ML/T^2), we need to analyze the components that contribute to force in this scenario.
In circular motion, the force exerted by the string is known as the centripetal force. It keeps the object moving in a circular path. Mathematically, the centripetal force (F) is given by:
F = (m * v^2) / r
Where:
- F is the force
- m is the mass of the object
- v is the speed of the object
- r is the radius of the circular path
Now, let's break down the dimensions of each variable in this equation:
- m (mass): The dimensions of mass are [M]
- v (speed): Speed is defined as distance divided by time (v = d/t). The dimensions of distance are [L] and the dimensions of time are [T].
Therefore, the dimensions of speed are [L/T].
In the equation, speed is squared (v^2), so the dimensions of speed squared become [(L/T)^2].
- r (radius): The dimensions of radius are [L].
By substituting these dimensions into the centripetal force equation:
F = (m * v^2) / r
F = [M] * [(L/T)^2] / [L]
To simplify, we can express this as:
F = [M * (L/T)^2 / L]
By rearranging the equation, we obtain the dimensions of force (F):
F = [ML/T^2]
Therefore, the correct combination of variables that gives the dimensions (ML/T^2) for the force in the given scenario is:
Force (F) = mass (m) * speed squared (v^2) / radius (r)