As you know, a common example of a harmonic oscillator is a mass attached to a spring. in this problem, we will consider a horizontally moving block attached to a spring. note that, since the gravitational potential energy is not changing in this case, it can be excluded from the calculations. for such a system, the potential energy is stored in the spring and is given byu=12kx2,where k is the force constant of the spring and x is the distance from the equilibrium position. the kinetic energy of the system is, as always, k=12mv2,where m is the mass of the block and v is the speed of the block. we will also assume that there are no resistive forces; that is, e=constant. a) find the total energy of the object at any point in its motion. b) find the amplitude of the motion. c) find the maximum speed attained by the object during its motion.

A)

Explanation:

The total energy of the system at any point in the motion is equal to the sum of the elastic potential energy of the spring, U, and of the kinetic energy of the mass, K:

where

k is the spring constant

x is the compression/stretching of the spring with respect to its equilibrium position

m is the mass of the block attached to the spring

v is the speed of the block

B)

Explanation:

The amplitude of the motion corresponds to the maximum displacement of the mass-spring system. The displacement of the system, x(t), at time t, for a simple harmonic oscillator is given by

where

A is the amplitude

is the angular frequency of the motion

t is the time

is the phase (we can take for simplicity)

The amplitude of the motion occurs when the displacement of the motion is maximum: x=A. In terms of energy, the mass-spring system is at its maximum displacement (x=A) when all the mechanical energy of the system is elastic potential energy, so when the kinetic energy is zero:

(1)

C)

The maximum speed of the system occurs when the elastic potential energy is zero: U=0 and the kinetic energy is maximum, so:

Due to the law of conservation of the mechanical energy, this energy must be equal to the energy of the system at its maximum displacement (1), so we can write

and solving for we find an expression for the maximum speed:

no. dot product   =0 when vecs at right angles

air exerts forces on falling objects near earth’s surface.

explanation:

the term 'free falls' refer to object that are falling towards the earth, in a situation in which gravity is the only force acting on the object.

for an object in free fall, the acceleration is constant (g, the gravitational acceleration), and so the velocity of the object constantly increases.

for objects near earth's surface, there is always another force acting on the object: the resistance due to the air. this force is opposed to the direction of fall of the object, and it is proportional to the velocity of the object, so as the object falls down, this force increases up to a point when it balances the force of gravity, and the object continues its motion at constant velocity (called terminal velocity). since in this case gravity is not the only force acting on the object, we are not in a situation of free fall.