Instantaneous center

When looking for the velocities of points on a rigid body, the method of instantaneous centers can often be used. If the velocity of two points on the body are known, those points and all other points on the body can be considered to be rotating with the same angular velocity about some motionless central point. This central point is called the instantaneous center of zero velocity. The instantaneous center generally moves through space as a function of time and has acceleration. It does not represent a point about which acceleration may be determined. 

If 0 is a fixed axis or the instantaneous center of zero velocity, then Equation 2-27 reduces to... 

In Figure 2-16 a 10 lb cylinder with a 3-in. radius rolls down a 30° incline. What is its angular acceleration and the linear acceleration of its center of mass In the free-body diagram of Figure 2-16, the point of contact between the wheel and the ramp is the instantaneous center of zero velocity. Thus,... 

Torque can be defined as the effectiveness of a force to produce rotation. It is the product of the force and the perpendicular distance from its line of action to the instantaneous center of rotation. 

The propagation of the wavepacket is thereby reduced to the solution of coupled first-order differential equations for the parameters representing the Gaussian wavepacket, with the true potential being expanded about the instantaneous center of the wavepacket [i2(<),f(<)]. This propagation scheme is very appealing and efficient provided the basic assumptions are fulfilled. The essential prerequisite is that the locally quadratic approximation of the PES is valid over the spread of the wavepacket. This rules out bifurcation of the wavepacket, resonance effects, or strong an-harmonicities. 

Centripetal force. The centripetal force is the radial component of the net force acting on a body when the problem is analyzed in an inertial system. The force is inward toward the instantaneous center of curvature of the path of the body. The size of the force is mv2/ where r is the instantaneous radius of curvature. See centrifugal force. 

The origin of the molecular system is chosen as the instantaneous center of mass. From the vector definitions above it therefore follows that... 

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically [36-38]. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as... 

Figure 3 is apart of Figure 2. When the vehicle moves fromp top, it revolves d9 around instantaneous center c at radius r. Radial lines cp and Cp intersect the reference path circle at P and P respectively. The angle between CP and CP is denoted d(j>...

When the atomic displacements from equilibrium are not large during the collision, it is sufficient to introduce equilibrium positions and displacements, r = d + u with the displacements defined as usual so that the instantaneous center-of-mass position and Euler angles = ( ", T ), =... 

Adding the equation of moments equilibrium with regards to the instantaneous center of rotation (point O), allow us to write the relation linking load P and tangential rolling force as a function of characteristic lengths of the contact area (b and d) ... 

---