Physics

Simple Harmonic Movement Time Functions (continued)


Hourly speed function

From the elongation time function at least two different paths can be followed to determine the time speed function. One is to use differential calculus and derive this equation as a function of time by obtaining an equation for velocity in the MHS.

Another way is to continue using the comparison with the MCU, remembering that, for circular motion, linear velocity is described as a vector tangent to the path:

Decomposing the tangential velocity vector:

Notice that the v It is negative because the vector is contrary to the elongation vector, so the movement is retrograde.

But we know that in an MCU:

and

So we can override these equalities and we have the hourly speed function in MHS:

Acceleration time function

Similarly to the hourly velocity function, the acceleration hourly function can be obtained using differential calculus by deriving velocity as a function of time. But it can also be calculated using the comparison with the MCU, remembering that when the motion is uniformly circular the only acceleration by which a body is subjected is that which causes it to change its meaning, ie the centripetal acceleration.

Decomposing the centripetal acceleration vector:

Notice that the The It is negative because the vector is contrary to the elongation vector, so the movement is retrograde.

But we know that in an MCU:

We can override these equalities and we will have the hourly acceleration function in MHS:

or

Some important observations:

  • The phase is always measured in radians.
  • The heartbeat can be defined by:

  • The early stage It is the same as the initial angle of motion in a trigonometric cycle, ie it is the sine wave lag angle.

For example, at time t = 0, a particle describing an MHS is in position , then its initial phase is determined by representing the given point projected on the trigonometric cycle:

Examples:

(1) A particle in MHS, with amplitude 0.5m, has a pulse equal to and early stage , what is your elongation, velocity and acceleration 2 seconds after the beginning of the movement?