Answer
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Hint
To find the relation between conservative force and the potential energy use the formula by assuming the object at two different points-
$W_{ab} = U_a - U_b$
Or simply it can be also said as “Work done is equal to the negative change in the potential energy of the particle”.
Complete step-by-step answer:
Considering an object whose mass is $m$ and placed at point ‘$a$’ at height ‘$h_1$’ and the work done required to move the point from another point ‘B’ at height ‘$h_2$’. As we know that potential at point ‘a’ is given by $mg(h_1)$ or “ $U_a$”, and potential energy at point ‘B’ is given by $mg(h_2)$ or “$U_b$”.
And hence the work done in moving the particle from point ‘a’ to ‘b’ is given by
$W_{ab} = U_a - U_b$,
Or it can be also written as-
$W_{ab} = -( U_b – U_a)$ ………………(1)
Or it can be also define as-
The force due to the conservative field $mg$ is downward and in the opposite direction to the direction in which the mass is move upward a distance $h$ so the work done by the force due to conservative field is ‘$ - mgh$’ and that by the definition is minus the change in the potential energy.
$-( U_b – U_a)$ can also be written in the form $ - \dfrac{{dU}}{{dx}}$
Hence equation (1) can also be written as-
$\vec F = - \dfrac{{dU}}{{dx}}$
Potential energy of the particle is equal to the negative of the work done by the conservative force.
Note
A conservative force is a force with the property that the total work done in moving a particle between the two points is independent of the path taken. Gravitational force is an example of the conservative force.
To find the relation between conservative force and the potential energy use the formula by assuming the object at two different points-
$W_{ab} = U_a - U_b$
Or simply it can be also said as “Work done is equal to the negative change in the potential energy of the particle”.
Complete step-by-step answer:
Considering an object whose mass is $m$ and placed at point ‘$a$’ at height ‘$h_1$’ and the work done required to move the point from another point ‘B’ at height ‘$h_2$’. As we know that potential at point ‘a’ is given by $mg(h_1)$ or “ $U_a$”, and potential energy at point ‘B’ is given by $mg(h_2)$ or “$U_b$”.
And hence the work done in moving the particle from point ‘a’ to ‘b’ is given by
$W_{ab} = U_a - U_b$,
Or it can be also written as-
$W_{ab} = -( U_b – U_a)$ ………………(1)
Or it can be also define as-
The force due to the conservative field $mg$ is downward and in the opposite direction to the direction in which the mass is move upward a distance $h$ so the work done by the force due to conservative field is ‘$ - mgh$’ and that by the definition is minus the change in the potential energy.
$-( U_b – U_a)$ can also be written in the form $ - \dfrac{{dU}}{{dx}}$
Hence equation (1) can also be written as-
$\vec F = - \dfrac{{dU}}{{dx}}$
Potential energy of the particle is equal to the negative of the work done by the conservative force.
Note
A conservative force is a force with the property that the total work done in moving a particle between the two points is independent of the path taken. Gravitational force is an example of the conservative force.
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