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 \(-1\).

Key concepts

Circular Motion

Circular motion occurs when an object moves along a path in a circular shape. This might sound a bit complex, but think of it like a roller coaster moving along a circular loop! 😊 One main feature here is that the speed of the object, known as its linear velocity, stays constant.
What's changing is the direction of the velocity as the object moves through different points on the circle. This curving motion creates a change in velocity, even if the speed doesn't change, which in turn provides what's called centripetal acceleration.
Key points to note about circular motion include:

• The object remains at a constant distance from the center or "pivot point" of its path.
• If the object's speed remains constant, the angular momentum stays conserved regarding points on the circle's circumference.
• This motion contrasts with linear motion where an object moves in a straight line without curving around a center.

The constant change in velocity direction while keeping a consistent speed is fundamental to understanding how forces act in circular paths.

Torque

Torque can be thought of as the measure of how much a force acting on an object causes that object to rotate. It's like the push or pull we apply to twist off a jar lid! 🤔 Torque is heavily reliant on two factors: the size of the force and the distance from the pivot point where the force is applied.
The formula to remember for torque is:
\[ \tau = r \times F \times \sin(\theta) \]
In this equation:

• \(\tau\) represents torque.
• \(r\) is the distance lever (or radius).
• \(F\) is the force applied.
• \(\theta\) is the angle between the force direction and the radius.

Effective torque occurs when the force is applied at a right angle to the radius, increasing the object's angular momentum. When there is no net torque, it means there is no change in rotational motion, conserving the system's angular momentum.

Conservation Laws

Conservation laws are essential principles in physics. They state how certain quantities don't change over time when isolated from other influences. Imagine having a bag of marbles; if you don't add or remove any marbles, the count remains the same. 🎮Angular momentum, like kinetic energy and mass, is conserved when outside forces don't interfere.
In the context of rotational motion, if no external torque (twisting force) acts on a system, the angular momentum remains constant. We use the equation:
\[ \tau = \frac{dL}{dt} \]
Here are some important points about conservation of angular momentum:

• If \(\tau = 0\), then \(\frac{dL}{dt}=0\), meaning \(\ L \) (angular momentum) is constant.
• This principle is used across many domains, from physics to engineering, ensuring the predictability of systems.
• An example can be seen in an ice skater spinning, where they pull in their arms to spin faster — their angular momentum is conserved, changing their speed as a balance to their arm movement.

Linear Velocity

Linear velocity is straightforward: it's the speed at which an object moves along a straight path. 🤗 In simpler terms, it's the distance an object travels over some time.
When objects move in circular paths, like a car turning around a track, their linear velocity is constantly changing direction. However, the speed might remain constant if the motion is uniform.
Some foundational ideas to know include:

• Linear velocity is calculated as \(v = \frac{d}{t}\) where \(d\) is the distance and \(t\) time.
• For circular motion, the linear velocity connects to the angular velocity by \(v = r \omega\), where \(\omega\) is the angular velocity and \(r\) is the radius.
• Even if the speed doesn't change, constant directional change influences the linear velocity.

Understanding linear velocity helps us break down how objects behave in both straight lines and curved paths, building the bridges between linear and rotational motions.