Introduction to physics

Lecture 21 Gyro and Statics Motion of a gyro Statics Virtual work

Moving a spinning wheel

A top

Gyroscopic precession The precession of a gyroscope shows up in many “common” situations.

Actual motion of a Gyro, nutation When we let go of a ryro, it first fall, and at the same time, process. It falls below the average procession line and then comes up again, and falls again in the vertical direction. The oscillation damps out eventually, one has in the end the pure procession.

Static equilibrium of rigid body Mechanism of Confinement Static equilibrium of rigid body Dynamical equations of rigid body Center of mass is at rest No rotation Mechanism of Confinement

Lever 𝐹 𝑙 1 𝐹− 𝑚 1 𝑔− 𝑚 2 𝑔=0 𝑚 1 𝑔 𝑙 1 − 𝑚 2 𝑔 𝑙 2 =0 𝑚 2 𝑔 𝑚 1 𝑔

View point of energy Energy conservation 𝐸=𝑇+𝑉=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 No kinetic energy 𝑇=0 When energy is 𝐸 1 or 𝐸 2 , the system is in the equilibrium state Condition of equilibrium is the potential energy is at local minimum (stable equilibrium) or maximum (unstable equilibrium)

Stable equilibrium

Unstable equilibrium

Shape of hanging bridge

Protein Structure Prediction

Mechanism of Confinement A elastic ring on a cone a h Mechanism of Confinement

Mechanism of Confinement Virtual displacement （虚位移）: displacement under constraint within the variation of time dt=0 Real displacement Virtual work（虚功）: Ideal constraint（理想约束）: Virtual displacement Mechanism of Confinement

Constraint forces 𝐹 𝑐𝑖 and other forces 𝐹 𝑖 Total virtual work 𝛿𝑊= Σ 𝑖 ( 𝐹 𝑖 + 𝐹 𝑐𝑖 ) ∙𝛿 𝑟 𝑖 Ideal constraint Σ 𝑖 𝐹 𝑐𝑖 ∙𝛿 𝑟 𝑖 =0 Static equilibrium Σ 𝑖 𝐹 𝑖 + 𝐹 𝑐𝑖 =0→ Σ 𝑖 𝐹 𝑖 ∙𝛿 𝑟 𝑖 =0

Mechanism of Confinement l1 l2 q1 q2 m1g m2g Mechanism of Confinement