# Derive the equations of motion for a rigid body undergoing planar motion

To derive the equations of motion for a rigid body undergoing planar motion, we need to consider the rotational and translational motion of the body. In planar motion, the body moves in a plane, and all particles of the body move in parallel planes.

Let's start by considering the translational motion of the rigid body. We assume that the body has a mass "m" and its center of mass moves in the xy-plane. The position vector of the center of mass is given by r = (x, y), where x and y are the coordinates of the center of mass.

Derive the equations of motion for a rigid body undergoing planar motion-To describe the translational motion, we need to consider the forces acting on the body. These forces can be divided into external forces and internal forces. External forces are those exerted on the body by external agents, such as gravity or applied forces. Internal forces are forces exerted between different particles within the body.

According to Newton's second law of motion, the sum of external forces acting on the body equals the mass of the body times its acceleration. Mathematically, this can be written as:

• ΣF_ext = m * a,

where ΣF_ext represents the sum of external forces, m is the mass of the body, and a is the acceleration of the center of mass.

Derive the equations of motion for a rigid body undergoing planar motion-Now, let's consider the rotational motion of the rigid body. We assume that the body can rotate about an axis perpendicular to the xy-plane. To describe the rotational motion, we need to consider the moments (or torques) acting on the body.

Similar to the translational motion, the sum of external moments acting on the body equals the moment of inertia of the body times its angular acceleration. Mathematically, this can be written as:

• ΣM_ext = I * α,

where ΣM_ext represents the sum of external moments, I is the moment of inertia of the body, and α is the angular acceleration.

To derive the equations of motion, we need to relate the translational and rotational quantities. This can be done using the parallel-axis theorem, which states that the moment of inertia of a body about any axis parallel to an axis through the center of mass is given by the moment of inertia about the center of mass plus the mass of the body times the square of the distance between the two axes.

Using the parallel-axis theorem, we can express the moment of inertia I as:

• I = I_cm + m * d^2,

Derive the equations of motion for a rigid body undergoing planar motion-where I_cm is the moment of inertia about the center of mass, m is the mass of the body, and d is the distance between the center of mass and the axis of rotation.

Now, let's derive the equations of motion. We start by differentiating the position vector r with respect to time:

• v = dr/dt = (dx/dt, dy/dt),

where v represents the velocity of the center of mass.

Next, we differentiate the velocity vector v with respect to time to obtain the acceleration vector a:

• a = dv/dt = (d²x/dt², d²y/dt²).

Derive the equations of motion for a rigid body undergoing planar motion-We can rewrite the acceleration vector a as the sum of two components: linear acceleration a_cm, which represents the acceleration of the center of mass, and angular acceleration α times the perpendicular distance d from the center of mass to the axis of rotation. Mathematically, this can be written as:

• a = a_cm + α * d.

Using these expressions for a and α, we can rewrite Newton's second law for translational motion as:

• ΣF_ext = m * a_cm,

where ΣF_ext represents the sum of external forces acting on the body.

Conclusion

We have derived the equations of motion for a rigid body undergoing planar motion. We started by considering the translational motion of the body and used Newton's second law to relate the sum of external forces to the mass and acceleration of the center of mass.

Derive the equations of motion for a rigid body undergoing planar motion-Next, we considered the rotational motion of the body and used the parallel-axis theorem to relate the moment of inertia about an axis parallel to the center of mass to the moment of inertia about the center of mass.

By differentiating the position vector, we obtained the velocity and acceleration vectors of the center of mass. We then expressed the acceleration vector as the sum of linear acceleration and angular acceleration multiplied by the perpendicular distance from the center of mass to the axis of rotation.

Derive the equations of motion for a rigid body undergoing planar motion-Finally, we rewrote Newton's second law for translational motion and the equation for rotational motion using the derived expressions for the acceleration and moment of inertia.

These equations of motion provide a mathematical framework for analyzing the motion of a rigid body undergoing planar motion. They allow us to predict the translational and rotational behavior of the body based on the forces and moments acting on it.

Derive the equations of motion for a rigid body undergoing planar motion-Understanding the equations of motion for planar motion is crucial in various fields, such as physics, engineering, and robotics, where the motion of rigid bodies plays a significant role. By applying these equations, we can analyze and design systems involving planar motion, enabling us to predict and control the behavior of complex mechanical systems.

## FAQ.

Q: What is planar motion?

A: Planar motion refers to the motion of a rigid body that occurs within a single plane. In planar motion, all particles of the body move in parallel planes, and their motion is restricted to this plane.

Q: What is the difference between translational and rotational motion?

A: Translational motion refers to the linear motion of a body where all particles move in the same direction and with the same velocity. Rotational motion, on the other hand, refers to the motion of a body around an axis, where particles move in circular paths around that axis.

Q: What is the center of mass?

A: The center of mass of a system or object is the point at which the entire mass of the system or object can be considered to be concentrated. For a rigid body, the center of mass is the point at which the body can be balanced.

Q: What is Newton's second law of motion?

A: Newton's second law of motion states that the net force acting on an object is equal to the product of its mass and acceleration. Mathematically, it can be written as ΣF = m * a, where ΣF represents the sum of forces, m is the mass of the object, and a is its acceleration.

Q: What is the parallel-axis theorem?

A: The parallel-axis theorem states that the moment of inertia of a body about any axis parallel to an axis through the center of mass is equal to the moment of inertia about the center of mass plus the mass of the body multiplied by the square of the distance between the two axes.