Question

For n-t coordinate, describe the direction of unit vector n and unit vector t How many scalar equations can we obtain from the equations of motion? What is the principle of work and energy? When is best to use it? How does one determine whether a force does work or not? What is the principle of impulse and momentum? What is the difference between linear momentum and angular momentum? When is momentum conserved typically?

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Answer 1

Normal/Tangential Coordinates

Sometimes it is useful to describe motion by giving the path that a particle is moving along and the speed of the particle at each point along the path.

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This coordinate system is convenient to use when particles move along a surface of known shape.

Coordinates are chosen at each point along the part such that:

e_t is a unit vector tangential to the path and pointing in the direction of motion, and

e_n is a unit vector normal to the path and pointing toward the center of curvature.

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where s is the position along the path.

Velocity of the particle has direction e_t (it is always tangent to the path) and a magnitude equal to the rate at which the particle moves along the path, such that

vp(t) = se; = vet

where v = ds/dt.

2-

Equations of Motion

In more complex cases (usually 3-D), a Cartesian vector is written for every force and a vector analysis is often best. Threescalar equations can be written from this vector equation. You may only need twoequations if the motion is in 2-D.

3-

The Work-Energy Theorem

The principle of work and kinetic energy (also known as the work-energy theorem) states that the work done by the sum of all forces acting on a particle equals the change in the kinetic energy of the particle. This definition can be extended to rigid bodies by defining the work of the torque and rotational kinetic energy.

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Kinetic Energy: A force does work on the block. The kinetic energy of the block increases as a result by the amount of work. This relationship is generalized in the work-energy theorem.

The work W done by the net force on a particle equals the change in the particle’s kinetic energy KE:

W=ΔKE=12mv2f−12mv2iW=ΔKE=12mvf2−12mvi2

where v_i and v_f are the speeds of the particle before and after the application of force, and mis the particle’s mass.

Derivation

For the sake of simplicity, we will consider the case in which the resultant force F is constant in both magnitude and direction and is parallel to the velocity of the particle. The particle is moving with constant acceleration a along a straight line. The relationship between the net force and the acceleration is given by the equation F = ma (Newton’s second law), and the particle’s displacement d, can be determined from the equation:

v2f=v2i+2advf2=vi2+2ad

obtaining,

d=v2f−v2i2ad=vf2−vi22a

The work of the net force is calculated as the product of its magnitude (F=ma) and the particle’s displacement. Substituting the above equations yields:

W=Fd=mav2f−v2i2a=12mv2f−12mv2i=KEf−KEi=ΔKE

4- a force doesn't do work when displacement is perpendicular to line of action of force in other case it does work.

5-The Principle of Impulse and Momentumdescribes how an object's linear and angular momentum change with applied forces and moments. ... It applies to both particle motion and rigid body motion.

6-Angular momentum is inertia of rotation motion. Linear momentum is inertia of translation motion. The big difference isthat the type of motion which is related to each momentum is different. It isimportant to consider the place where the force related to rotation applies, which isappears as 'r' in the formula.

7- momentum is conserved typically when there is no impulsive force act externally on the system