A particle is constrained to move on a parabola $y = kx^{2}$. The number of degrees of freedom is:

Question

The Lagrangian of a system is given by $L = \tfrac{1}{2}\dot{q}^{2} + q\dot{q} - \tfrac{1}{2}q^{2}$. It describes the motion of:

Answer 1

Step 1: Parametrize the allowed positions. Every point on the parabola can be written using one parameter, say $x=s$, giving $y=ks^{2}$.

Step 2: Since a single parameter $s$ pins down the location completely, the configuration space is one-dimensional.

Step 3: The number of degrees of freedom equals the dimension of the configuration space, which is $1$.

Step 4: Equivalently, starting from a planar particle (2 coordinates) and removing one holonomic constraint leaves $2-1=1$.

Step 5: Hence the motion has
\[\boxed{f = 1}\]
degree of freedom.

A particle is constrained to move on a parabola \(y = kx^{2}\). The number of degrees of freedom is:

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The Correct Option is A

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