A test machine that kicks soccer balls has a 5-lb simulated foot attached to the end of a 6-ft long pendulum arm of negligible m

Answer:

Explanation:

The half-life for the specified RC circuit can be expressed as

where [/tex]\tau = RC[/tex]

Given

The circuit has a resistance of 40 ohms, and by adding a new resistor of 48 ohms, the total resistance becomes 40 + 48 = 88 ohms.

Thus, the new half-life is

Now, divide equation 2 by 1

After substituting all values, we can calculate the revised half-life

Answer:

Explanation:

The equilibrium vacancy concentration can be described by:

nv/N = exp(-ΔHv/KT),

where T is the temperature at which vacancies form,

K = Boltzmann's constant,

and ΔHv = enthalpy of vacancy formation.

Rearranging this equation to express temperature allows you to calculate it using the provided values. A detailed breakdown of the process is included in the attached file.

Answer:

Time constant = 15.34 seconds

The thermometer indicates an error margin of 0.838°

Explanation:

Given

t = 1 minute = 60 seconds

c(t) = 98% = 0.98

According to the details provided, the thermometer functions as a first-order system.

The transfer function of such a system is expressed as;

C(s)/R(s) = 1/(sT + 1).

To determine the time constant, the step response must be evaluated.

This is defined as

r(t) = u(t) --- Applying Laplace Transformation

R(s) = 1/s

Replacing 1/s back into C(s)/R(s) = 1/(sT + 1).

What we have

C(s)/1/s = 1/(sT + 1)

C(s) = 1/(sT + 1) * 1/s

C(s) = 1/s - 1/(s + 1/T) --- Taking Inverse Laplace Transformation

L^-1(C(s)) = L^-1(1/s - 1/(s + 1/T))

Given that e^-t <–> 1/(s + 1) --- {L}

1 <–> 1/s {L}

Thus, the unit response c(t) = 1 - e^-(t/T)

Substituting 0.98 for c(t) and 60 for t

0.98 = 1 - e^-(60/T)

0.98 - 1 = - e^-(60/T)

-0.02 = - e^-(60/T)

e^-(60/T) = 0.02

ln(e^-(60/T)) = ln(0.02)

-60/T = -3.912

T = -60/-3.912

T = 15.34 seconds

It follows that the time constant = 15.34 seconds

The error signal is characterized by

E(s) = R(s) - C(s)

Where the temperature shifts at 10°/min; which equals 10°/60 s = 1/6

Thus,

E(s) = R(s) - 1/6 C(s)

Calculating C(s)

C(s) = 1/s - 1/(s + 1/T)

C(s) = 1/s - 1/(s + 1/15.34)

Note that R(s) = 1/s

Thus, E(s) turns into

E(s) = 1/s - 1/6(1/s - 1/(s + 1/15.34))

E(s) = 1/s - 1/6(1/s - 1/(s + 0.0652)

E(s) = 1/s - 1/6s + 1/(6(s+0.0652))

E(s) = 5/6s + 1/(6(s+0.0652))

E(s) = 0.833/s + 1/(6(s+0.0652)) ---- Taking Inverse Laplace Transformation

e(t) = 1/6e^-0.652t + 0.833

In a first-order system, a steady state condition is reached when the time is four times the time constant.

Thus,

Time = 4 * 15.34

Time = 61.36 seconds

Consequently, e(t) becomes

e(t) = 1/6e^-0.652t + 0.833

e(t) = 1/(6e^-0.652(61.36)) + 0.833

e(t) = 0.83821342824942664566211

e(t) = 0.838 --- Rounded off

Thus, the thermometer reveals an error of 0.838°

The increase in temperature of the helium gas is calculated to be 14.25 K. The helium is located in an insulated box that falls from a height of 4.5 km. As it descends, the potential energy is transformed into internal energy of the helium gas. The equation for temperature change can be expressed as: 10 x 4.5 = 3.15 x ΔT, yielding a temperature increase of ΔT = 14.25 K.

Answer:

Explanation:

In a steady-state condition, the total volume exiting matches the total volume entering, and

the total amount of salt entering equals the total amount of salt exiting.

The volume entering each minute is = 2000 + 2 x 10³ x 60

= 122000 L.

The total salt entering per minute = 1200 x 2000 + 20 x 2 x 10³ x 60

= 2400000 + 2400000 mg

= 4800000 mg.

The volume of water exiting each minute = 122000 L.

The total salt exiting each minute = 4800000 mg.

The concentration of salt in the exiting water = 4800000 / 122000

= 39.344 mg / L.