9.5 · Apply introductory planar rigid-body equations of motion

University of Alberta EN PH 131: Engineering Mechanics: Dynamics

Introduction to Rigid-Body Dynamics

Connecting external forces and moments to translation and rotation

What you will learn

• Describe planar rigid-body motion using the mass center's acceleration and the body's angular acceleration.
• Draw a free-body diagram and choose consistent positive directions.
• Apply force and moment equations to find unknown accelerations or forces.
• Use the no-slip relation for a body rolling on a fixed surface.

1. Describe the motion and draw the forces

• Identify the body, observer, reference frame, and axes.
• Show external forces and their application points.
• Choose positive rotation before assigning moment signs.

2. Apply the planar equations of motion

• Use force equations for mass-center translation.
• Use moments about $G$ for angular acceleration, or use a suitable fixed point with its matching inertia.
• Unknown reaction forces are included as external forces.

3. Add constraints, solve, and check

• Use only justified motion constraints and keep their signs consistent.
• Check units, directions, and limiting cases.
• A negative result reports direction relative to your chosen positive direction.

Worked example

A bar released from a pin

1. Set the system and signs
   Take the bar as the system, viewed from the stationary ground frame. Let counterclockwise be positive. At release, the bar has zero angular velocity, but it can have nonzero angular acceleration.
2. Take moments about the pin
   The pin reactions have zero moment about the pin. The weight acts downward at distance $L/2$, creating a clockwise, negative moment. For rotation about a fixed pin, use the bar's inertia about that pin.
   $$-mg\frac{L}{2}=I_P\alpha,\qquad I_P=\frac{1}{3}mL^2$$
3. Solve and interpret
   The negative result means the bar initially rotates clockwise. The mass cancels, as expected for a uniform bar under its own weight.
   $$\alpha=-\frac{3g}{2L}=-12.3\,\mathrm{rad/s^2}$$

Worked example

A force applied off the mass center

1. Define the system and axes
   Take the plate as the system in the stationary ground frame. Let $+x$ point right, $+y$ point up, and counterclockwise rotation be positive. The upward guide force balances weight, so the net vertical force is zero.
2. Find translation
   The only unbalanced force is the $12\,\mathrm{N}$ horizontal force. Apply the force equation to the mass center.
   $$a_{Gx}=\frac{12}{4.0}=3.0\,\mathrm{m/s^2},\qquad a_{Gy}=0$$
3. Find rotation
   The horizontal force line is $0.20\,\mathrm{m}$ below $G$, so it creates a counterclockwise positive moment. For a thin rectangular plate, use its planar mass moment of inertia about $G$.
   $$I_G=\frac{m}{12}(w^2+h^2)=0.173\,\mathrm{kg\,m^2},\qquad \alpha=\frac{(12)(0.20)}{I_G}=13.8\,\mathrm{rad/s^2}$$

Worked example

A solid cylinder rolling down an incline

1. Choose incline-aligned coordinates
   Take the cylinder as the system in the ground frame. Let positive translation point down the incline and positive rotation be clockwise, the direction that accompanies rolling downhill. The initial speed is zero; find the accelerations and contact friction.
2. Write translation and rotation equations
   Along the incline, gravity pulls downhill and static friction acts uphill. The normal force acts at the contact point along the radius toward the center, so its line of action passes through the center and it has no moment about $G$. Use the solid-cylinder inertia and the no-slip acceleration relation.
   $$mg\sin 30^\circ-f=ma,\qquad fR=I_G\alpha,\qquad a=R\alpha,\qquad I_G=\frac{1}{2}mR^2$$
3. Solve for acceleration and friction
   Substituting the rolling relation into the moment equation gives the friction needed for rotation. Combining it with translation yields the acceleration and then the friction magnitude.
   $$a=\frac{2}{3}g\sin 30^\circ=3.27\,\mathrm{m/s^2},\qquad f=mg\sin 30^\circ-ma=3.27\,\mathrm{N}$$

Common mistakes and how to avoid them

Lesson summary

• For planar rigid-body motion, net force determines mass-center acceleration and net moment determines angular acceleration.
• Draw the body, external forces, force locations, axes, and positive rotation direction before writing equations.
• Use an inertia value that matches the moment reference point and include only justified constraints.
• Check units, signs, physical direction, and limiting cases.

Check your understanding

Key terms

Mass center

The point used to represent a body's translation in the force equation.

Mass moment of inertia

A measure of a body's resistance to angular acceleration about a specified axis.

Angular acceleration

The rate of change of angular velocity, measured here in radians per second squared.

Rolling without slipping

Rolling contact with no relative motion between the contacting surfaces at the instant of contact.

Continue through EN PH 131

View the complete EN PH 131 University of Alberta EN PH 131: Engineering Mechanics: Dynamics curriculum and lessons

• 9.1 · Distinguish translation and rotation about a fixed axis
• 9.2 · Relate angular position, velocity, and acceleration
• 9.3 · Relate velocities of points on a planar rigid body
• 9.4 · Relate accelerations of points on a planar rigid body
• 1.2 · Relate position, path length, and displacement
• 1.3 · Differentiate position to obtain velocity and acceleration

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