A particle moves in a potential region given | vedclass

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Question

(C) The force $\vec{F}$ is related to the potential energy $U$ by the negative gradient: $\vec{F} = -
abla U = -\left( \hat{i} \frac{\partial U}{\par

Vedclass · Accepted Answer

$-2k(x\hat{i} + y\hat{j} + z\hat{k})$

Vedclass · Answer

(C) The force $\vec{F}$ is related to the potential energy $U$ by the negative gradient: $\vec{F} = -
abla U = -\left( \hat{i} \frac{\partial U}{\partial x} + \hat{j} \frac{\partial U}{\partial y} + \hat{k} \frac{\partial U}{\partial z} ight)$.Given $U = K(x^2 + y^2 + z^2)$.Calculating the partial derivatives:$\frac{\partial U}{\partial x} = 2Kx$,$\frac{\partial U}{\partial y} = 2Ky$,and $\frac{\partial U}{\partial z} = 2Kz$.Substituting these into the force equation:$\vec{F} = -(2Kx\hat{i} + 2Ky\hat{j} + 2Kz\hat{k})$.Therefore,$\vec{F} = -2K(x\hat{i} + y\hat{j} + z\hat{k})$.

Column-$I$	Column-$II$
$(1)$ Conservative force	$(a)$ Frictional force
$(2)$ Non-conservative force	$(b)$ Gravitational force
	$(c)$ Internal force

$A$ particle moves in a potential region given by the formula $U = K(x^2 + y^2 + z^2)$. Find the force acting on the particle.

Similar Questions

The potential energy of an object of mass $m$ moving in the $xy$ plane in a conservative field is given by $U = ax + by$,where $x$ and $y$ are position coordinates of the object. Find the magnitude of its acceleration.

Obtain the relation between potential energy and force for a conservative force.

Match Column-$I$ with Column-$II$.

Column-$I$ Column-$II$

$(1)$ Conservative force $(a)$ Frictional force

$(2)$ Non-conservative force $(b)$ Gravitational force

$(c)$ Internal force

Which quantity is obtained by taking the negative derivative of potential energy with respect to displacement in the case of a conservative force?

Give three definitions of conservative force.

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