**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.

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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.

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