How can I analyze an equation and identify its relationship types?

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To analyze the equation T_s=2π√(m/k) and identify the relationships between T_s, m, and k, it is essential to understand the forms of proportionality. T_s is directly proportional to the square root of m, indicating a radical relationship, while it is inversely proportional to the square root of k, suggesting an inverse relationship. Simplifying the equation is not necessary to identify these relationships, as the existing form already reveals the nature of the connections. Understanding these types of relationships is crucial for interpreting the equation's implications in physics. This analysis helps clarify how changes in mass and spring constant affect the period T_s.
AznBoi
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Ok I'm having trouble analyzing this equation from Physics:

T_s=2\pi\sqrt\frac{m}{k}
 
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woah that was my first time doing Latex with those variables lol. Anyways, how do I identify the direct, inverse, radical, etc. relationships between the two sides of the equal sign? The 2pi is multiplied by \sqrt\frac{m}{k} and I don't know how to describe the relation ships between Ts and m, Ts and k, etc. Do I have to simplify the equation or get rid of the sqrt?
 
AznBoi said:
woah that was my first time doing Latex with those variables lol. Anyways, how do I identify the direct, inverse, radical, etc. relationships between the two sides of the equal sign? The 2pi is multiplied by \sqrt\frac{m}{k} and I don't know how to describe the relation ships between Ts and m, Ts and k, etc. Do I have to simplify the equation or get rid of the sqrt?
A direct relationship is of the form y = kx
You would say y is directly proportional to x
An inverse relationship is of the form y = k/x
You would say y is inversely porportional to x

y could also be directly or inversely proportional to a power or root of x.

We call the universal gravition law an inverse square law because the force in inversely proportional to the square of the distance between objects.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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