A jump rope could be crudely modeled by the equation putt = Tuxx u (0,t) = sin (wt) U (1, t) = sin (wt) where p is the rope density, T is the tension, and w is the forcing frequency. In general there are initial conditions associated with the problem as well, but in this problem we will only be interested in the particular solution. We will assume that damping causes the initial conditions to be forgotten (however, we will *not* include the damping explicitly, as it makes the algebra much messier). (a) Split the problem into two parts u (x, t) = ub (x, t) +u+ (x, t), and solve the boundary problem by direct integration. (b) Identify the interior problem, noting that the time-dependent boundary conditions of the boundary problem produce an interior problem with a non-homogeneous forcing. (c) Solve the interior problem by eigenfunction expansion. [Note: When you have reduced your problem to an ODE by means of the transform, you may ignore the homogeneous solutions (set both integration constants to zero), because we are ignoring the initial conditions.] Now, the goal of a jump rope is to get the whole rope to oscillate more or less as a unit. That is, we want the shape of the solution to be dominated by the first mode sin (TX), without lots of wiggles due to the subsequent terms in the expansion. (d) Note that the coefficients of your eigenfunction expansion depend on {p, T, w}. For a jump rope, p is likely to be fixed. Why? On the other hand, t and w are likely to be adjustable. Why? (e) Find a simple relation between 1 and w that induces resonance in the first mode sin (7x), and thus optimizes the jump rope experience.

Do you need to draw this out?

Infrared sensor is most used sensor in wireless technology where remote controlling functions and detection of surrounding objects/ obstacles are involved. this post will discuss about what is infrared sensor, its working principle, how it works, types, applications, advantages and disadvantages. what is infrared sensor

ir sensor is a simple electronic device which emits and detects ir radiation in order to find out certain objects/obstacles in its range. some of its features are heat and motion sensing.

ir sensors use infrared radiation of wavelength between 0.75 to 1000µm which falls between visible and microwave regions of electromagnetic spectrum. ir region is not visible to human eyes. infrared spectrum is categorized into three regions based on its wavelength i.e. near infrared, mid infrared, far infrared.

wavelength regions of infrared spectrum

near ir – 0.75µm to 3 µm
mid ir – 3 µm to 6 µm
far ir – > 6 µm
1 introduction to infrared sensor

fig. 1 – introduction to infrared sensor

working principle of infrared sensor

infrared sensors works on three fundamental physics laws:

planck’s radiation law: any object whose temperature is not equal to absolute zero (0 kelvin) emits radiation.
stephan boltzmann law: the total energy emitted at all wavelengths by a black body is related to the absolute temperature.
wein’s displacement law: objects of different temperature emit spectra that peak at different wavelengths that is inversely proportional to temperature.
key elements of infrared detection system

the key elements of infrared detection system are:

ir transmitter

ir transmitter acts as source for ir radiation. according to plank’s radiation law, every object is a source of ir radiation at temp t above 0 kelvin. in most cases black body radiators, tungsten lamps, silicon carbide, infrared lasers, leds of infrared wavelength are used as sources.

transmission medium

as the name suggests, transmission medium provides passage for the radiation to reach from ir transmitter to ir receiver. vacuum, atmosphere and optical fibers are used as medium.

ir receiver

generally ir receivers are photo diode and photo transistors. they are capable of detecting infrared radiation. hence ir receiver is also called as ir detector. variety of receivers are available based on wavelength, voltage and package.

ir transmitter and receivers are selected with matching parameters. some of deciding specifications of receivers are photosensitivity or responsivity, noise equivalent power and detectivity.