System of Particles and Rotational Motion · Lesson 15 of 15
Chapter Summary and Practice
“Connect centre-of-mass motion, equilibrium and rotation through model choice and mixed practice.”
• Choose the correct system, reference origin and axis. • Connect particle sums to whole-body laws. • Distinguish force, torque and conservation conditions. • Apply formulas with their assumptions and units.
A connected description of an extended body
Start by deciding whether deformation is negligible. Choose a system boundary, frame, origin and rotation axis. The CM describes overall translation; angular variables describe orientation and rotation. For CM acceleration sum external forces. For angular-momentum change sum external torques about the same reference. Internal forces can redistribute energy and motion even when they cancel from the total-force sum. To describe both translation and rotation, one set of variables is not enough.
| Question | Main relation | Condition or caution |
|---|---|---|
| Where is CM? | R = Σmᵢrᵢ/M | Use signed coordinates and actual mass weights |
| How does CM accelerate? | MA = F_ext | Constant-mass particle system |
| Is linear momentum conserved? | dP/dt = F_ext | Check resultant force or relevant component |
| How large is torque? | τ = Fd⊥ | State origin and perpendicular lever arm |
| How does angular momentum change? | dL/dt = τ_ext | Central internal forces cancel pair torques |
| Is a supported body at rest in equilibrium? | ΣF = 0 and Στ = 0 | Include weights, reactions and friction |
| How does a fixed-axis rotor accelerate? | τ_z = Iα | Constant I; otherwise d(Iω)/dt |
| What is rotational energy? | K_rot = Iω²/2 | Use the same axis as I |
| How does internal redistribution change spin? | I_iω_i = I_fω_f | Negligible external axial torque |
Connections that prevent common mistakes
Mass tells how much material a body contains; I also records where the material is relative to an axis. CM and CG coincide in uniform gravity but express different ideas. Force and torque equilibrium are independent conditions. Torque and energy have equal dimensions yet different meanings. Linear speed differs between points of a rotor even though angular velocity is common. Angular momentum can remain constant while angular speed and kinetic energy change. These distinctions connect the chapter’s many formulas.
For an equilibrium calculation, draw the force diagram first. For a cross product, fix the order and right-hand direction. For a rotation problem, convert revolutions to radians and name the inertia axis. For a conservation calculation, include the complete system and identify the missing external force or torque component. A numerical answer is incomplete if its sign, axis or physical interpretation is unclear.
Practice Problems
- Locate CM for masses 2 kg at x = 0 and 6 kg at x = 4 m. Explain the inverse division of distances.
- A uniform L-plate consists of three equal 1 m squares. Derive its CM by component centres.
- A uniform disc loses a radius-R/2 hole centred R/2 to the right. Find the remaining CM and explain the sign.
- Derive MA = F_ext by differentiating the CM definition and explain which Newton laws are used.
- Use a = (3, −4, 5) and b = (−2, 1, −3) to find the dot product, cross product and area of their spanning triangle.
- Show that a · (b × c) gives signed parallelepiped volume and explain the absolute value for volume.
- Derive the torque law for a particle and show why central internal forces cancel in a system.
- Explain how zero resultant force can coexist with nonzero torque, and give a couple example.
- Find reactions for the 70 cm loaded bar and the ladder discussed earlier; identify every moment arm.
- Use all three equilibrium equations for the angled-string suspended bar. Explain why its CG is 0.72 m from the left end.
- Derive I and rotational energy for a thin ring and a pair of equal end masses on a light rod.
- Convert 1200 rpm and 3120 rpm to SI angular speeds and obtain the revolutions during the 16 s acceleration interval.
- Solve the cord-driven flywheel by torque, work–energy and angular kinematics; compare results.
- For a system changing from I to 2I/5 without axial torque, find angular-speed and kinetic-energy ratios. Explain the energy source.
- Explain why L_z = Iω remains useful even when the full L vector is not parallel to ω.
Selected answers and reasoning
Problem
Check the first and vector-product problems.
- 1.The two-mass CM is 6(4)/8 = 3 m from the 2 kg mass, so its distances from the masses are 3 m and 1 m.
- 2.For the L-plate the two coordinates are each 5/6 m. The hole problem gives X = −R/6 and Y = 0.
- 3.The vector dot product is −25 and cross product is (7, −1, −5). Its magnitude is √75 = 5√3, so triangle area is 5√3/2 in the appropriate squared edge-length units.
Problem
Connect the numerical applications.
- 1.Beam reactions are 54.88 N and 43.12 N. For the ladder, wall reaction and floor friction are each 34.65 N and floor normal is 196 N.
- 2.The motor-wheel angular acceleration is 4π rad/s² and its angular displacement is 576 revolutions.
- 3.The 20 kg, 0.20 m flywheel has I = 0.40 kg m², α = 12.5 rad/s² and final rotational energy 50 J after 2 m of cord unwinds.
- 4.Reducing total I to 2I/5 multiplies angular speed and kinetic energy by 5/2. Angular momentum stays fixed and internal work supplies the energy rise.
Quiz
Which pair of equilibrium conditions is required for static rigid-body balance?
Static equilibrium requires both translational and rotational balance.
Which conservation condition is correct?
The torque law specifies the conservation condition.
What must accompany a tabulated I value?
Different axes through the same body can give different I.
Which body may have CM in empty space?
The ring’s symmetry places its CM at its empty centre.
If a driving torque balances friction at steady ω, what is true?
The engine can supply energy dissipated by resistance without changing ω.
Key Takeaways
• Separate CM translation from motion relative to CM. • Reference origin, axis and system boundary belong in the solution. • Force, torque and energy relations answer different questions. • Use conservation and standard formulas only with their stated conditions.
Previous · Lesson 14
Fixed-Axis Angular Momentum and Its Conservation
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