A bead of mass m$\mathrm{m}$ slides without friction on the wall of a vertical circular hoop of radius R$\mathrm{R}$ as shown in figure. The bead moves under the combined action of gravity and a massless spring (k) attached to the bottom of the hoop. The equilibrium length of the spring is R$\mathrm{R}$ . If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes R$\mathrm{R}$ , would be (spring constant is k$\mathrm{k}$ , g is acceleration due to gravity)

Core Logic

Assertion A: Central force configurations depend solely on the positional distance parameter r$\mathrm{r}$. They are strictly conservative fields, making path metrics completely irrelevant for total work evaluation. (True)

Reason R: Non-conservative profiles like friction or drag dissipate thermal energy paths and do not possess any state potential energy function. (True)

Since statement R provides general information about non-conservative forces rather than stating why central fields are path independent, it fails as a direct explanatory bridge.

Step 1: Final Conclusion

Both assertions are factually accurate, but R is not the appropriate explanation for A. This selects option (2).

Pattern Recognition

Conservative fields allow defining potential profiles (F = -∇ U$F = -\nabla \mathrm{U}$); non-conservative structures completely break this relation.

Chapter Mix

Class 11 Physics: Work, Energy and Power

Related Formula

Conservation of Mechanical Energy:

At the initial height S$S$ (velocity v = 0$v = 0$):

At any height x$x$ above the ground:

Core Logic

Let the height of the particle at that instant be x$x$. We are given: K = 3U$K = 3U$

Substitute the energy terms:

By energy conservation:

Step 1: Calculating the Speed

Now find the speed v$v$ at this height x = S/4$x = S/4$. Since K = 3U$K = 3U$:

Pattern Recognition

Standard ratio trick: If K = n U$K = n U$, then by energy conservation (n+1)U = Etotal$(n+1)U = E_{\text{total}}$. This immediately yields:

Here n = 3$n = 3$, so x = S/4$x = S/4$. This rapid shortcut lets you find the height in a split second!

Chapter Mix

Class 11 Physics: Work, Energy and Power

Core Logic

Potential energy functions are strictly mathematically defined exclusively for conservative force interactions via the relationship F = -(dU)/(dx)$F = -\frac{dU}{dx}$[cite: 727]. Coulomb's force, Gravitational force, and Spring restoring force are completely path-independent conservative fields[cite: 146, 147, 149]. Frictional force is a path-dependent, dissipative non-conservative force[cite: 148, 727, 728]. Consequently, it is impossible to define a scalar potential energy function for mechanical friction[cite: 727, 728].

Pattern Recognition

Whenever you encounter a potential energy definition requirement, remember that it is a direct marker for conservative fields. Dissipative forces like friction or viscous drag instantly break this condition[cite: 727, 728].

Chapter Mix

Class 11 Physics: Work, Energy and Power

Related Formula

F = m · a$F = m \cdot a$ [cite: 778]

Core Logic

Given velocity as a function of position x$x$: [cite: 164, 779]

v = 4√(x) v² = 16x$$v = 4\sqrt{x} \implies v^2 = 16x$$ [cite: 164, 779]

Differentiating both sides with respect to position coordinate x$x$: [cite: 780]

2v(dv)/(dx) = 16 v(dv)/(dx) = 8$$2v\frac{dv}{dx} = 16 \implies v\frac{dv}{dx} = 8$$ [cite: 780, 781]

Thus, the acceleration of the object is a constant value a = 8 m/s²$a = 8\ \text{m/s}^2$[cite: 781]. Converting mass to kilograms (m = 500 g = 0.5 kg$m = 500\ \text{g} = 0.5\ \text{kg}$) [cite: 164, 788]:

F = 0.5 × 8 = 4 N$$F = 0.5 \times 8 = 4\ \text{N}$$ [cite: 788]

Pattern Recognition

When velocity depends on position coordinate x$x$ like v = k√(x)$v = k\sqrt{x}$, squaring instantly reveals that acceleration is constant, since v² = k² x$v^2 = k^2 x$ matches the third kinematic profile v² = 2ax$v^2 = 2ax$ directly[cite: 779].

Chapter Mix

Class 11 Physics: Work, Energy and Power

Related Formula

The work done W$W$ by a variable force component F(x)$F(x)$ over a displacement step is given by:

Core Logic

Assuming the object moves from the origin x=0$x=0$ to x=1 m$x=1\text{ m}$ :

Step 1: Integration and Variable Isolation

Perform the integration step :

Given α = 1 N$\alpha = 1\text{ N}$ , substitute this into the equation to find β$\beta$ :

Pattern Recognition

For polynomial forces, the integration steps always yield fractional coefficients matching their power index (1$1$ for constant, (1)/(3)$\frac{1}{3}$ for squared terms).

Chapter Mix

Class 11 Physics: Work, Energy and Power