FE Review Mechanical Session 5 (Dynamics)

Summary

The session reviews dynamics topics from the FE Reference Handbook and works through several representative problems. The presenter emphasizes recognizing which relationship applies and using shortcuts—such as conservation laws or instantaneous centers—instead of lengthy derivations.

Key Concepts Reviewed

• Particle kinematics: Velocity is the time derivative of position; acceleration is the time derivative of velocity, or the second derivative of position. Dot notation represents time derivatives.
• Motion descriptions: The handbook covers rectilinear and curvilinear motion, radial and transverse components, relative motion, and circular motion. For circular paths, arc length relates to angle by (s = r\theta), with (\theta) in radians.
• Constant-acceleration and projectile motion: Standard kinematic equations apply when acceleration is constant. Projectile motion can be split into horizontal and vertical components.
• Particle kinetics: Newton’s second law relates net force to acceleration: (\sum F = ma). The presenter contrasts this with statics, where the net force is zero.
• Energy and work: Energy methods relate kinetic and potential energy between states. Work by nonconservative forces must also be included when applicable. Rigid-body kinetic energy includes both translational and rotational parts; spring energy is (\tfrac12 kx^2), and gravitational potential energy is (mgh).
• Other handbook topics: The review briefly points to power, efficiency, impulse and momentum, friction, rigid-body equations of motion, mass moments of inertia, the parallel-axis theorem, radius of gyration, angular momentum, and vibration. Vibrations are not covered in the worked problems, though the presenter notes they may warrant additional study.

Worked Problems and Methods

1. Initial velocity from a position function

   • Differentiate the position function to obtain velocity.
   • Evaluate the velocity at (t = 0) to find the initial velocity.
   • The result in the example is (20).

2. Projectile flight time

   • Resolve the initial velocity into vertical and horizontal components.
   • Use vertical motion: the vertical velocity is zero at the projectile’s peak.
   • With a launch speed of (110\ \text{m/s}) and an angle of (20^\circ), the time to the peak is about (3.84) seconds.
   • With level launch and landing and no air resistance, the descent takes the same amount of time, giving a total flight time of about (7.67) seconds.

3. Block sliding down a frictional incline

   • Choose an axis along the incline and resolve gravity into components parallel and perpendicular to it.
   • The normal force determines friction: (F_f = \mu N).
   • Apply (\sum F = ma) along the incline to find the acceleration.
   • Use (v^2 = v_0^2 + 2a\Delta x). For the example’s 52 kg block, the presenter obtains a final speed of about (9.83\ \text{m/s}).

4. Banked curve without relying on tire friction

   • Interpret the question as asking for a bank angle that supplies the needed centripetal acceleration without friction.
   • Use the speed and curve radius; the vehicle’s mass and the stated friction coefficient are not needed for this idealized calculation.
   • Convert (100\ \text{km/h}) to metres per second and use the banking relation (\tan\theta = v^2/(rg)).
   • For a 200 m radius, the angle is approximately (21^\circ).

5. Spinning skater and conservation of angular momentum

   • When the skater draws in her arms, her mass moment of inertia decreases.
   • If external torque is negligible, angular momentum remains constant; therefore, angular velocity increases.
   • The presenter treats the related decrease in radius of gyration as consistent with the reduced moment of inertia and selects “all of the above.”

6. Linkage angular velocity using an instantaneous center

   • Locate the instantaneous center from the geometry and directions of motion.
   • Find the known point’s linear speed using (v = r\omega).
   • Relate that speed to the other link through its distance from the same instantaneous center, then use (v = r\omega) again to obtain the unknown angular velocity.
   • The example uses the linkage’s geometry, including a 3–4–5 triangle, to determine the relevant distance.

7. Spring compression and energy conservation

   • Since the mass barely reaches the specified point and the guide is frictionless, equate the initial spring energy to the final gravitational potential energy.
   • Set (\tfrac12 kx^2 = mgh) and solve for the compression (x).
   • The presenter reports approximately (1.35), rounded to (1.4); the units and full input values are unclear in the subtitles.

8. Two-block linkage and instantaneous-center velocity relation

   • Extend the blocks’ motion directions to locate the instantaneous center.
   • Use the linkage geometry to find the distances from the center to each block.
   • Equate the angular speed of the rigid linkage as viewed from either point, giving a velocity ratio based on those distances.
   • With the stated geometry, the presenter calculates distances of roughly (2.6\ \text{m}) and (1.5\ \text{m}). From (v_A = 3\ \text{m/s}), this gives (v_B \approx 5.2\ \text{m/s}).

The presenter closes by encouraging use of the FE handbook, recognition of efficient solution paths, and seeking help through the session’s Discord community.

Speakers and Sources

• Primary presenter: Leads the FE dynamics review; not clearly named in the subtitles.
• Cameron: An attendee addressed by the presenter who responds during the discussion.
• Other attendees: Briefly answer questions and participate; their identities are unclear in the subtitles. One attendee mentions needing to leave for a meeting.
• Source/reference: The FE Reference Handbook, used for the topic overview and equations.
• Briefly mentioned source: “Whistler’s class” is referenced by an attendee; the context is unclear.