Plane Motion of Rigid Bodies: Forces and Accelerations

Why it matters

Understanding the plane motion of rigid bodies and the forces and accelerations involved is crucial for designing and analyzing mechanical systems such as machinery, vehicles, and structural components. This knowledge helps engineers predict how these systems will behave under various forces, ensuring safety and efficiency.

Key ideas

• Plane Motion of Rigid Bodies: This involves the movement of bodies in a two-dimensional plane, where each point in the body moves in parallel planes.
• Types of Motion: Rigid bodies can undergo translational motion, rotational motion, or a combination of both (general plane motion).
• Newton's Second Law for Rigid Bodies: The sum of external forces and moments acting on a rigid body is equal to the mass times the acceleration of the center of mass and the moment of inertia times the angular acceleration, respectively.
• Equations of Motion: These are derived from Newton's laws and are used to solve problems involving forces and accelerations in rigid bodies.

Equations of planar motion

For a constant-mass rigid body in an inertial frame, let G be the centre of mass. The planar equations are:

ΣF_x = m a_Gx, ΣF_y = m a_Gy, ΣM_G = I_G α.

ΣF is the resultant of all external forces, ΣM_G is the resultant external moment about G, and I_G is the mass moment of inertia about the axis through G perpendicular to the motion plane. A force contributes r × F and a couple contributes its signed moment. Use a consistent positive rotational direction.

The simplified equation ΣM_O = I_O α is also valid for rotation about a fixed pivot O. It is not a general formula about an arbitrary accelerating point. When in doubt, take moments about G.

For another point A fixed in the body:

a_G = a_A + α × r_G/A + ω × (ω × r_G/A).

The last term is the centripetal acceleration. For planar motion it points from G toward A with magnitude ω²r_G/A. It exists even when α = 0.

Draw a free-body diagram to obtain forces and moments, then apply kinematic constraints separately. Rolling without slip, for example, relates centre acceleration and angular acceleration when rolling on a stationary straight surface; friction must still satisfy the available contact force.

Worked example

Problem: A rigid body with a mass of 10 kg has a net external force of 50 N directed along +x and a net counterclockwise external moment of 20 N·m about its centre of mass G. Its mass moment of inertia I_G is 5 kg·m². Assume planar motion. Calculate the linear and angular accelerations.

Given:

• Mass, m = 10 kg
• Force, F = 50 N
• Moment, M = 20 N·m
• Mass moment of inertia about G, I_G = 5 kg·m²

1. Calculate the linear acceleration using F = m·a:

   • a = F / m = 50 N / 10 kg = 5 m/s²

2. Calculate the angular acceleration using M = I·α:

   • α = M / I = 20 N·m / 5 kg·m² = 4 rad/s²

Answer: The linear acceleration is 5 m/s² along +x and the angular acceleration is 4 rad/s² counterclockwise.

Common mistakes

• Confusing linear and angular quantities, such as using mass instead of moment of inertia for rotational motion.
• Forgetting to convert units, especially when dealing with moments and forces.
• Neglecting the effect of distributed forces and moments on the motion of the body.

For GATE ME

Questions often involve calculating the linear and angular accelerations of rigid bodies under various force and moment conditions. Practice problems that require setting up and solving equations of motion for different types of plane motion.

Quick check

1. What is the formula for calculating the moment of a force?
2. How do you calculate the angular acceleration of a rigid body?
3. What is the relationship between force and linear acceleration?

Answers: 1. M = F·d (where d is the perpendicular distance from the axis of rotation), 2. α = ΣM_G / I_G for planar motion, 3. ΣF = m·a_G.