The potential energy of a particle of mass 5 kg moving in the x-y plane is given by U=-7 x+24 y

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The potential energy of a particle of mass 5 kg moving in...

The potential energy of a particle of mass $5 \mathrm{~kg}$ moving in the $x-y$ plane is given by $U=-7 x+24 y$ joule, $x$ and $y$ being in metre. Initially at $t=0$ the particle is at the origin and moving with velocity $(2 \hat{i}+3 \hat{j}) \mathrm{m} / \mathrm{s}$. The magnitude of the force on the particle and the acceleration of the particle are (a) $35 \mathrm{~N}, 7 \mathrm{~m} / \mathrm{s}^{2}$ (b) $30 \mathrm{~N}, 6 \mathrm{~m} / \mathrm{s}^{2}$ (c) $25 \mathrm{~N}, 5 \mathrm{~m} / \mathrm{s}^{2}$ (d) None of these

The potential energy of a particle of mass $5 \mathrm{~kg}$ moving in the $x-y$ plane is given by $U=-7 x+24 y$ joule, $x$ and $y$ being in metre. Initially at $t=0$ the particle is at the origin and moving with velocity $(2 \hat{i}+3 \hat{j}) \mathrm{m} / \mathrm{s}$. The magnitude of the force on the particle and the acceleration of the particle are
(a) $35 \mathrm{~N}, 7 \mathrm{~m} / \mathrm{s}^{2}$
(b) $30 \mathrm{~N}, 6 \mathrm{~m} / \mathrm{s}^{2}$
(c) $25 \mathrm{~N}, 5 \mathrm{~m} / \mathrm{s}^{2}$
(d) None of these

Key Concepts

Gradient of a Scalar Field

The gradient of a scalar function is a vector that points in the direction of the greatest rate of increase of the function, with its magnitude representing the rate of change. In the context of potential energy, calculating the partial derivatives with respect to spatial coordinates yields the components of the force vector (after applying a negative sign).

Newton's Second Law

Newton's Second Law states that the net force acting on a particle is equal to the product of its mass and its acceleration (F = ma). This principle is used to relate the force derived from the potential energy to the resulting acceleration of the particle.

Potential Energy

Potential energy is a scalar quantity that represents the energy stored in a system due to its position or configuration. In conservative force fields, changes in potential energy are associated with the work done by or against the force field.

Force from Potential Energy (Conservative Force)

In a conservative force field, the force acting on a particle is given by the negative gradient of the potential energy function. This negative gradient indicates that the force points in the direction of decreasing potential energy.

Vector Magnitude

The magnitude of a vector, such as force or velocity, is computed as the square root of the sum of the squares of its components. Understanding how to calculate vector magnitudes is essential for determining the overall strength of the force or the speed of an object.

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