AIIMS 2005 Physics Conservation Laws Assertion Reason Question

To tackle the question regarding the conservation of angular momentum in a central force field, let's break down the assertion and the reason, and analyze their validity.

Assertion

For a system of particles under central force field, the total angular momentum is conserved.

A central force field is defined such that the force acting on a particle points towards a fixed point (the center) and only depends on the distance from that center. Mathematically, a central force can be expressed as:

$\vec{F} = f(r) \hat{r}$

where $r$ is the distance from the center and $\hat{r}$ is the unit vector pointing radially outward from the center.

For a system of particles, the total angular momentum $\vec{L}$ about the center is given by:

$\vec{L} = \sum_{i} \vec{r}_i \times \vec{p}_i$

where $\vec{r}_i$ is the position vector of the $i$-th particle and $\vec{p}_i$ is its linear momentum. In a central force field, since the force does not exert any torque about the center of force, the rate of change of angular momentum is given by:

$\frac{d\vec{L}}{dt} = \vec{\tau}$

where $\vec{\tau}$ is the torque. If the torque is zero, angular momentum is conserved:

$\frac{d\vec{L}}{dt} = 0 \implies \vec{L} \text{ is constant}$

Thus, the assertion is true.

Reason

The torque acting on such a system is zero.

In a central force field, the torque $\vec{\tau}$ about the center is defined as:

$\vec{\tau} = \vec{r} \times \vec{F}$

Since the force $\vec{F}$ is radial (directed along $\hat{r}$), the position vector $\vec{r}$ and the force $\vec{F}$ are parallel. The cross product of two parallel vectors is zero:

$\vec{\tau} = \vec{r} \times f(r) \hat{r} = 0$

Therefore, the reason is also true.

Conclusion

Since both the assertion and the reason are true, and the reason correctly explains the assertion (as the torque being zero leads to the conservation of angular momentum), the correct answer is:

A) Both assertion and reason are true and reason is the correct explanation of assertion.

Why Other Options are Incorrect

• Option B: Both assertion and reason are true, but the reason is not the correct explanation. This is incorrect because the reason directly explains why the assertion is true.

• Option C: Assertion is true, but reason is false. This is incorrect since both the assertion and reason are indeed true.

• Option D: Both assertion and reason are false. This option is false as both statements are true.

In summary, the conservation of angular momentum in a central force field is a fundamental concept in mechanics, underscored by the relationship between torque and angular momentum.