Question

Identify the correct statement for the rotational motion of a rigid body \(\\{A\\}\) Individual particles of the body do not undergo accelerated motion \\{B \\} The centre of mass of the body remains unchanged. \\{C\\} The centre of mass of the body moves uniformly in a circular path \\{D\\} Individual particle and centre of mass of the body undergo an accelerated motion.

Short answer

The correct statement for the rotational motion of a rigid body is \(D: \textrm{Individual particle and centre of mass of the body undergo an accelerated motion}\).

Step-by-step solution

Identify the Key Concepts

The first step to solve this exercise is to recall the basics of rotational motion of a rigid body. A rigid body is an object with a shape that does not change. In rotational motion, every particle of the body moves in a circular path about a line. This line, also called the axis of rotation, penetrates the body. Contrary to translational motion, particles in a rigid body do not all have the same velocity or acceleration.

Evaluate Statement A

For a rigid body in rotational motion, all particles move about the axis in circular paths. This means that these particles are undergoing an accelerated motion, given the constant change of the motion direction. Remember that acceleration is defined as any change in the velocity of an object, which includes both magnitude and direction changes. Therefore, statement A is incorrect.

Evaluate Statement B

The center of mass of a rigid body in rotation does not necessarily remain unchanged. It depends on the type of rotation. In cases of pure rotation, the center of mass is fixed, but in cases of rolling or general plane motion, for example, the center of mass can change its position over time. Therefore, statement B can be true in some circumstances, but not in all. It is a partially correct statement.

Evaluate Statement C

Regarding statement C, It is also not a universally true statement. The center of mass of a rotating body does not always move uniformly in a circular path. As before, this depends on the specific type of rotational motion in question. Thus, statement C is also partially correct.

Evaluate Statement D

Finally, let's consider statement D. In rotational motion, every particle of the rigid body indeed moves in a circular path about a line, meaning both the individual particle and the centre of mass of the body undergo an accelerated motion, because their direction is continuously changing. Therefore, statement D is correct. So, the correct statement for the rotational motion of a rigid body is "D: Individual particle and centre of mass of the body undergo an accelerated motion."

Key concepts

Rigid Body Dynamics

When we discuss rigid body dynamics, we're focusing on how solid objects move. A rigid body is simply an object with a fixed shape that does not deform, no matter how much force is applied. This makes it easier to predict and analyze movement because the relative positions of all particles within the body remain constant. The study of these motions is essential for understanding everything from spinning tops to rolling cars.

A key feature of rigid body dynamics is that it involves both translational and rotational motion.

• In **translational motion**, all parts of the body move in the same direction and speed, like a moving car on a straight road.
• In **rotational motion**, every part of the body moves in a circular path around an axis. For instance, the wheels of a car as it travels down the street.

Although particles in a rigid body can have different velocities and accelerations, they move around a fixed axis, resulting in a coordinated motion. Understanding these dynamics helps us explain and predict the paths and behaviors of various objects under different forces.

Center of Mass

The center of mass is a crucial concept in understanding both the translational and rotational motion of rigid bodies. It is defined as the average position of all mass in a body. You can think of it as the point at which the whole mass of the body is concentrated. This hypothetical point helps simplify the analysis of motion.

In cases of pure rotation, the center of mass itself can remain stationary while the entire body spins around it. However, in more complex motions—like rolling down an incline—the center of mass will have a path over time. Despite this, the center of mass allows us to accurately predict how a body behaves under forces.

• **For balanced systems**, the center of mass aligns with the pivot point, making calculations easier.
• **In everyday applications**, knowing the center of mass helps design stable structures, vehicles, and even clothing.

By finding the center of mass, we can effectively reduce a problem involving the motion of multiple particles into a simpler form, using a single point's motion.

Circular Motion

Circular motion is fundamental to rotational dynamics, involving any motion where an object rotates along a circular path. It can be uniform, where the speed is constant, or non-uniform, where the speed changes. In rigid body dynamics, every particle of the object participating in circular motion experiences centrifugal and centripetal forces.

Components of circular motion include:

• **Centripetal force**, which acts toward the center of the circular path, keeping the object in its orbit.
• **Centrifugal tendency**, which is not an actual force but a perceived effect where there appears to be an outward pull on every particle.
• The rotation speed influences the amount of these forces experienced by the body.

Understanding these forces and how they apply to each particle within the rigid body is crucial. These concepts explain behaviors like why satellites orbit planets or why a twirling baton remains in motion.