Question

If the resulting external force acting on system is zero then of the system is constant and if the resultant external torque acting on a system is zero then \(\ldots \ldots \ldots \ldots \ldots\) of the system is constant. (a) total energy, angular momentum (b) liner momentum, angular momentum (c) liner momentum, energy (d) angular and linear momentum

Short answer

\( \) Based on the analysis, if the external force is zero, the linear momentum (P) will be conserved. And if the external torque is zero, the angular momentum (L) will be conserved. So, the short answer is (b) linear momentum, angular momentum.

Step-by-step solution

Understand Newton's second law and external force

When the external force acting on a system is zero, it means that the net force acting on the system is zero. According to Newton's second law, this implies constant linear momentum for the system, as: \[ \sum F = \frac{dP}{dt} .\]

Understand the relationship between torque and angular momentum

When the external torque acting on a system is zero, it means that the net torque acting on the system is also zero. Following this, and according to the conservation of angular momentum, the angular momentum of the system remains constant: \[ \sum \tau = \frac{dL}{dt} .\]

Choose the correct option based on the analysis

From the analysis in steps 1 and 2, we can see that if the external force is zero, the linear momentum (P) will be conserved. And if the external torque is zero, the angular momentum (L) will be conserved. Thus, the correct answer is: (b) linear momentum, angular momentum

Key concepts

Linear Momentum

In physics, linear momentum is a fundamental concept that describes the quantity of motion an object possesses. It is determined by the product of an object's mass and velocity. The formula used to calculate linear momentum is given by:

• Linear Momentum ( P ) = Mass ( m ) \( \times \) Velocity ( v ).

Linear momentum is a vector quantity. This means it has both magnitude and direction, making it important in evaluating collisions and movements.
In an isolated system, where there are no external forces acting, the total linear momentum remains constant. This principle is known as the conservation of linear momentum. It follows from Newton's second law, which is expressed as:

• \( \sum F = \frac{dP}{dt} \) .

Here, \( F \) represents the sum of forces, and \( \frac{dP}{dt} \) represents the rate of change of momentum. If the net external force is zero, then the derivative, \( \frac{dP}{dt} \), is zero, indicating that momentum does not change.
A real-world application of the conservation of linear momentum is evident in car collisions, where the total momentum of the involved vehicles before the crash is equal to the total momentum after the impact. This assumes no external forces to be acting.

Angular Momentum

Angular momentum, much like linear momentum, is concerned with the quantity of rotational motion. It is particularly important in the study of rotating objects such as wheels, planets, or any object moving along a circular path. The angular momentum of an object is determined by its moment of inertia (a measure of how much mass is distributed with respect to an axis) and its angular velocity. It is represented by the formula:

• Angular Momentum ( L ) = Moment of Inertia ( I ) \( \times \) Angular Velocity ( \( \omega \) ).

The conservation of angular momentum states that when no external torque acts upon a system, the total angular momentum remains constant. This is represented by:

• \( \sum \tau = \frac{dL}{dt} \) .

Here, \( \sum \tau \) stands for the sum of torques, and \( \frac{dL}{dt} \) represents the rate of change of angular momentum.
A common example of angular momentum in action is an ice skater spinning. When a skater pulls their arms closer to their body, the moment of inertia decreases, and as a result, their angular velocity increases to conserve angular momentum, making them spin faster.

Torque

Torque is often described as the rotational equivalent of force. It measures how effectively a force can twist an object around a pivot point or axis. Torque (\( \tau \)) is determined by the equation:

• Torque ( \( \tau \) ) = Force ( F ) \( \times \) Lever Arm Distance ( r ).

The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force.
Just like force in linear motion, torque causes changes in the rotational motion of objects. When multiple torques are acting, their vector sum decides the resultant torque, affecting the angular acceleration of an object.
If the net external torque is zero, the system's angular momentum remains constant, as dictated by:

• \( \sum \tau = \frac{dL}{dt} \) = 0.

An example of torque at work is using a wrench to turn a bolt. The force applied by hand translates through the wrench's length to exert torque on the bolt.