Examples of characteristic time constant in the following topics:
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- It takes time to build up energy, and it also takes time to deplete energy; hence, there is an opposition to rapid change.
- The initial current is zero and approaches I0=V/R with a characteristic time constant for an RL circuit, given by:
- The current will be 0.632 of the remainder in the next time.
- A well-known property of the exponential function is that the final value is never exactly reached, but 0.632 of the remainder to that value is achieved in every characteristic time $\tau$.
- The characteristic time $\tau$ depends on only two factors, the inductance L and the resistance R.
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- An object moving with constant velocity must have a constant speed in a constant direction.
- If an object is moving at constant velocity, the graph of distance vs. time ($x$ vs.
- $t$) shows the same change in position over each interval of time.
- You can also obtain an object's velocity if you know its trace over time.
- When an object is moving with constant velocity, it does not change direction nor speed and therefore is represented as a straight line when graphed as distance over time.
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- Constant acceleration occurs when an object's velocity changes by an equal amount in every equal time period.
- An object experiencing constant acceleration has a velocity that increases or decreases by an equal amount for any constant period of time.
- It is defined as the first time derivative of velocity (so the second derivative of position with respect to time):
- When it is not, we can either consider it in separate parts of constant acceleration or use an average acceleration over a period of time.
- Due to the algebraic properties of constant acceleration, there are kinematic equations that relate displacement, initial velocity, final velocity, acceleration, and time.
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- A black body in thermal equilibrium (i.e. at a constant temperature) emits electromagnetic radiation called black body radiation.
- Black body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature.
- where $B$ is the spectral radiance of the surface of the black body, $T$ is its absolute temperature, $\lambda$ is wavelength of the radiation, $k_B$ is the Boltzmann constant, $h$ is the Planck constant, and $c$ is the speed of light.
- It is not a surprise that he introduced Planck constant $h = 6.626 \times 10^{-34} J \cdot s$ for the first time in his derivation of the Planck's law.
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- The constant of proportionality is called the spring constant and is usually denoted by k.
- So this is its natural or characteristic frequency.
- Let's continue to refer to this characteristic frequency as $\omega_0$ to emphasize the fact that it is a constant for a given spring/mass system.
- So $B$ must equal whatever velocity the mass has when it zips through the origin, divided by the characteristic frequency $\omega_0$.
- Since the $t$ disappears, we see that the energy is constant with time, and thus energy is conserved.
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- An ideal gas has different specific heat capacities under constant volume or constant pressure conditions.
- where the partial derivatives are taken at: constant volume and constant number of particles, and at constant pressure and constant number of particles, respectively.
- The heat capacity ratio or adiabatic index is the ratio of the heat capacity at constant pressure to heat capacity at constant volume.
- It is a simple equation relating the heat capacities under constant temperature and under constant pressure.
- In addition, molecules in the gas may pick up many characteristic internal vibrations.
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- Imagine a particle moving in a circle around a point at a constant speed.
- At any instant in time, the particle is moving in a particular straight-line direction with that speed.
- Thus, while the object moves in a circle at constant speed, it undergoes constant linear acceleration to keep it moving in a circle.
- However, it's angular velocity is constant since it continually sweeps out a constant arc length per unit time.
- Constant angular velocity in a circle is known as uniform circular motion.
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- If the sources are constant (DC) sources, the result is a DC circuit.
- A direct current circuit is an electrical circuit that consists of any combination of constant voltage sources, constant current sources, and resistors.
- In this case, the circuit voltages and currents are independent of time.
- The solution to these equations usually contain a time varying or transient part as well as constant or steady state part.
- The two Kirchoff laws along with the current-voltage characteristic (I-V curve) of each electrical element completely describe a circuit.
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- Constant angular acceleration describes the relationships among angular velocity, angle of rotation, and time.
- For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotating through many revolutions.
- We have already studied kinematic equations governing linear motion under constant acceleration:
- Similarly, the kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
- As in linear kinematics where we assumed a is constant, here we assume that angular acceleration α is a constant, and can use the relation: $a=r\alpha $ Where r - radius of curve.Similarly, we have the following relationships between linear and angular values: $v=r\omega \\x=r\theta $
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- Isobaric processis a thermodynamic process in which the pressure stays constant (at constant pressure, work done by a gas is $P \Delta V$).
- An isobaric process is a thermodynamic process in which pressure stays constant: ΔP = 0.
- Because the change in volume of a cylinder is its cross-sectional area A times the displacement d, we see that Ad=ΔV, the change in volume.
- Specific heat at constant pressure is defined by the following equation:
- A graph of pressure versus volume for a constant-pressure, or isobaric process.