Question

Statement \(-1-\) The angular momentum of a particle moving in a circular orbit with a constant speed remains conserved about any point on the circumference of the circle. Statement \(-2\) - If no net torque outs, the angular momentum of a system is conserved. \(\\{\mathrm{A}\\}\) Statement \(-1\) is correct (true), Statement \(-2\) is true and Statement- 2 is correct explanation for Statement \(-1\) \\{B \\} Statement \(-1\) is true, statement \(-2\) is true but statement- 2 is not the correct explanation four statement \(-1\). \(\\{\mathrm{C}\\}\) Statement \(-1\) is true, statement- 2 is false \\{D \(\\}\) Statement- 2 is false, statement \(-2\) is true

Short answer

The correct answer is Option B. This is because both Statement 1 (The angular momentum of a particle moving in a circular orbit with a constant speed remains conserved about any point on the circumference of the circle) and Statement 2 (If no net torque acts, the angular momentum of a system is conserved) are true. However, Statement 2 is not precisely the correct explanation for Statement 1. In Statement 1, the conservation of angular momentum occurs due to the constant speed of the particle in circular motion, not specifically because of the absence of net torque as stated in Statement 2.

Step-by-step solution

Verify the conservation of angular momentum for a particle in circular motion

To verify statement 1, we can recall the equation for angular momentum (L) which is given by the formula: \[ L = mvr \cos(\theta) \] Here m is the mass of the particle, v is its linear velocity, r is the distance from the pivot point (or the center of the circular orbit), and θ is the angle between the vectors v and r. Since the problem states that the particle is moving in a circular orbit with constant speed, we assume that the linear velocity v is constant. Also, the radius of the circle, r, remains constant. As the particle moves around the circle, θ changes, but since we are only looking at points on the circumference, the angle θ is always 90 degrees. Thus, \(\cos(90^{\circ}) = 0\) and the product with \(mvr\) remains constant. This confirms that the angular momentum of the particle remains conserved about any point on the circumference of the circle. #Step 2: Verify Statement 2#

Verify the conservation of angular momentum when no net torque is acting

For Statement 2, we need to establish a relationship between net torque and angular momentum. We can refer to the equation: \[ \tau = \frac{dL}{dt} \] Where τ is the net torque and L is the angular momentum of the system. This equation states that the net torque is equal to the rate of change of angular momentum with respect to time. If there is no net torque acting on the system (\( \tau = 0\)), then the right-hand side of the equation (\(\frac{dL}{dt}\)) should also equal zero. This implies that the angular momentum (L) is constant over time, which confirms that if no net torque acts, the angular momentum of a system is conserved. #Step 3: Determine the relationship between statements 1 and 2#

Compare Statements 1 and 2

Since we have verified that both Statement 1 and Statement 2 are true, we now need to examine whether Statement 2 is the correct explanation for Statement 1. In Statement 1, it was established that the angular momentum of a particle moving in a circular orbit with a constant speed remains conserved about any point on the circumference of the circle. Meanwhile, Statement 2 states that if no net torque acts, the angular momentum of a system is conserved. The conservation of angular momentum in Statement 1 is observed due to the geometry of the problem (i.e., the constant speed of the particle in circular motion), and not directly related to the absence of net torque as mentioned in Statement 2. Therefore, although both statements are true, Statement 2 is not the correct explanation for Statement 1. Based on our analysis, the correct answer is: \(\\{\mathrm{B}\\}\) Statement \(-1\) is true, statement \(-2\) is true but statement- 2 is not the correct explanation four statement \(-1\).