Ch 6-7 Systems

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Questions and Answers

What is the mathematical description of the relation between the number of electrons (charge Q) and current i?

$Q(t) = \int i , dt + Q_0$

What is the unit of charge?

The unit of charge is the coulomb (C).

What is the unit of current?

The unit of current is the ampere (A).

What is the unit of voltage?

The unit of voltage is the volt (V). Signup and view all the answers

What is the formula for power dissipated in a linear resistor?

$P = i^2R = v^2R$ Signup and view all the answers

What is Kirchhoff's current law?

Conservation of charge Signup and view all the answers

What is Kirchhoff's voltage law?

Conservation of energy Signup and view all the answers

What is the series resistance law?

$vs = (R1 + R2)i$ Signup and view all the answers

What is the model for the capacitor voltage v1 in the series RC circuit?

\frac{dv1},{dt} + v1 = vs Signup and view all the answers

What is the expression for the capacitor voltage v1 (t) in the series RC circuit with a rectangular pulse input?

The expression for the capacitor voltage v1 (t) is not provided in the given text. Signup and view all the answers

What is the initial condition for the capacitor voltage v1 (0) in the series RC circuit with a rectangular pulse input?

The initial condition for the capacitor voltage v1 (0) is v1 (0) = 0. Signup and view all the answers

What is the current in the circuit?

$i = \frac{vs}{R} = \frac{6}{10} = 0.6$ A Signup and view all the answers

What is the power in the circuit?

$P = \frac{vs^2}{R} = \frac{6^2}{10} = 3.6$ W Signup and view all the answers

Derive the expression for $v3$ as a function of $v1$ and $v2$.

$v3 = 0.4(v1 + v2)$ Signup and view all the answers

Obtain the voltage $vo$ as a function of the applied voltage $vs$.

$vo = \frac{40}{9}vs + \frac{17}{2}$ Signup and view all the answers

What is the equation for the current-divider rule?

$i_1 = \rac{R_2},{R_1+R_2}i$ Signup and view all the answers

What is the equation for the parallel resistance law for two resistors?

$\rac{1},{R} = \rac{1},{R_1} + \rac{1},{R_2}$ Signup and view all the answers

What is the equation for the voltage-current relation of a diode?

$i = 0.16(e^{0.12v} - 1)$ Signup and view all the answers

What is the equation for the energy stored in a capacitor?

$E = \rac{1},{2}Cv^2$ Signup and view all the answers

What are Kirchhoff's current and voltage laws?

Kirchhoff's current law states that the total current entering a junction is equal to the total current leaving the junction. Kirchhoff's voltage law states that in any closed loop in a circuit, the sum of the voltages across all components is equal to the sum of the voltage sources. Signup and view all the answers

What is impedance and how is it related to electrical resistance?

Impedance is a generalization of the electrical resistance concept. It takes into account both the resistance and reactance (the effect of capacitors and inductors) in an electrical circuit. Impedance is represented by the symbol $Z$ and is related to resistance ($R$) and reactance ($X$) as $Z = R + jX$, where $j$ is the imaginary unit. Signup and view all the answers

What are the principles of direct-current (dc) motors?

The principles of direct-current (dc) motors involve the interaction between a magnetic field and an electric current to produce rotational motion. The motor converts electrical energy into mechanical energy. The direction of the motor's rotation is determined by the direction of the current flow in the motor's windings. Signup and view all the answers

What is the power consumed by the load in terms of vS?

PL = RL * (vL^2 / RL) = vL^2 / (RS + RL)^2 Signup and view all the answers

What is the maximum value of the ratio r?

The maximum value of the ratio r occurs when dr/dRL = 0. Therefore, r is maximized when RL = RS. Signup and view all the answers

What is the input-output relation for vo as a function of vi?

vo = G * vi / (1 + (GR2 / (R1 + R2))) Signup and view all the answers

What are the solutions for the currents i1, i2, and i3 in terms of the given voltages and resistances?

i1 = (R2 + R3)v1 - R2v2 / (R1R2 + R1R3 + R2R3), i2 = R3v1 + R1v2 / (R1R2 + R1R3 + R2R3), i3 = R2v1 - (R1 + R2)v2 / (R1R2 + R1R3 + R2R3) Signup and view all the answers

What is the equation for mass flow rate?

$qm = ho qv$ Signup and view all the answers

What is the unit of mass flow rate in FPS units?

slug/sec Signup and view all the answers

What is the unit of mass flow rate in SI units?

kg/s Signup and view all the answers

What is the equation for hydrostatic pressure at the bottom of a column of fluid?

$ ho gh$ Signup and view all the answers

What is the model for the motion of the displacement of the mass in part (a) of the figure?

$m ẍ = ( p1 - p2 )A$ Signup and view all the answers

What is the expression for the velocity of the displacement as a function of time?

$ẋ(t) = ẋ(0) + A m \int_{0}^{t} ( p1 (u) - p2 (u)) du$ Signup and view all the answers

What is the expression for the mass flow rate that must be delivered or absorbed by the two pumps?

If $ẋ > 0$, the pump providing pressure p1 must supply fluid at the mass rate $ρ A ẋ$, and the pump providing pressure p2 must absorb fluid at the same mass rate. Signup and view all the answers

What is the expression for the equivalent mass of the rack, pinion, and load?

$m_e = m + I R^2$ Signup and view all the answers

What is the difference between hydraulics and pneumatics?

Hydraulics is the study of systems in which the fluid is incompressible, while pneumatics is the study of systems in which the fluid is compressible. Signup and view all the answers

What is the conservation of mass principle?

The conservation of mass principle states that mass is neither created nor destroyed in a system, but it can change its form. Signup and view all the answers

What is the appropriate resistance relation for convection heat transfer?

The appropriate resistance relation for convection heat transfer is given by Newton's law of cooling, which states that the rate of heat transfer is proportional to the temperature difference between the object and its surroundings. Signup and view all the answers

What is the linearized model corresponding to the relation (7.3.7)?

The linearized model is given by the equation $R_r = 2B\frac{p_r}{q_m}$. Signup and view all the answers

How can the relation (7.3.8) be expressed in terms of $p_r$?

The relation can be expressed as $R_r = 2Bp_r$. Signup and view all the answers

What are the combination rules for series and parallel fluid resistances?

Series fluid resistances carry the same flow rate, while parallel fluid resistances have the same pressure difference across them. Signup and view all the answers

What is the equation for the fluid mass stored in the container?

m = ρA(x)dx Signup and view all the answers

What are the alternative hydraulic models of a container of fluid?

The alternative hydraulic models of a container of fluid are dm = qmi - qmo dt, C dm dp = qmi - qmo dt, and ρA dh = qmi - qmo dt Signup and view all the answers

What is the equation for the capacitance of a V-shaped trough?

C = \frac{2L \tan(\theta) \rho g}{p} Signup and view all the answers

What is the linearized resistance in terms of pressure and mass flow rate?

p = R_r qm Signup and view all the answers

Obtain the expression for the force $f_3$ in terms of the force $f_1$.

$f_3 = \rac{f_1 A_2},{A_1}$ Signup and view all the answers

What is the relation between the pressures $p_1$ and $p_2$ in the hydraulic brake system?

$p_1 = p_2 + ho gh$ Signup and view all the answers

Explain the tradeoff for force amplification in the hydraulic brake system.

The master piston must move a distance greater than that of the slave piston. Signup and view all the answers

What is the equation for conservation of mass in terms of mass inflow rate and mass outflow rate?

$ṁ = q_{mi} - q_{mo}$ Signup and view all the answers

What is the analogy between fluid mass accumulation and electrical capacitance?

The analogy between fluid mass accumulation and electrical capacitance is that fluid mass can accumulate within a system just like how a capacitor can store charge in an electrical circuit. Signup and view all the answers

What is the analogy between Kirchhoff's voltage law and the compatibility law in fluid systems?

The analogy between Kirchhoff's voltage law and the compatibility law in fluid systems is that both laws state that the sum of signed pressure differences (compatibility law) or voltage differences (Kirchhoff's voltage law) around a closed loop must be zero. Signup and view all the answers

What is the relation between fluid capacitance and the change in pressure?

The relation between fluid capacitance and the change in pressure is given by the equation $C = \frac{dm}{dp}$, where $C$ is the fluid capacitance, $dm$ is the change in stored mass, and $dp$ is the change in pressure. Signup and view all the answers

How can the expression for the capacitance of a storage tank be derived?

The expression for the capacitance of a storage tank can be derived by relating the liquid height $h$ to the liquid mass $m$ using $m = \rho Ah$, where $\rho$ is the density and $A$ is the cross-sectional area. The pressure due to the stored fluid mass can be expressed as $p = \rho gh$, where $g$ is the acceleration due to gravity. By differentiating the pressure with respect to the mass, the expression for the capacitance of the tank can be obtained as $C = \frac{dm}{dp} = \frac{A}{g}$. Signup and view all the answers

Newton’s first law states that a particle originally at rest, or moving in a straight line with a constant speed, will remain that way as long as it is not acted upon by an unbalanced external ______.

force Signup and view all the answers

The third law is summarized by the commonly used statement that every action is opposed by an equal ______.

reaction Signup and view all the answers

For an object treated as a particle of mass m, the second law can be expressed as dv = ______

f Signup and view all the answers

Conservation of mechanical energy is a direct consequence of ______'s second law.

Newton's Signup and view all the answers

The term on the left-hand side of the equal sign is called the ______ energy (KE).

kinetic Signup and view all the answers

If the work done by f (x) is independent of the path and depends only on the end points, then the force f (x) is derivable from a function V (x) as follows: dV (3.1.5) f (x) = − dx Then, in this case, f (x) is called a ______ force.

conservative Signup and view all the answers

This equation shows that V (x) has the same units as ______ energy.

kinetic Signup and view all the answers

According to Newton's law for mass m1, the equation of motion is given by m1ẍ = m1g - T1, where T1 is the ________ force in the cable.

tension Signup and view all the answers

The equation of motion for mass m2 is m2ÿ = T2 - m2g. T2 is the ________ force in the cable.

tension Signup and view all the answers

The equation of motion for mass m2 can be expressed in terms of the inertia I as Iθ̈ = RT1 - RT2 = R(T1 - T2). This equation represents the ________ law for mass m2.

Newton's Signup and view all the answers

The equation of motion for the pulley system is ẍI = (m1 - m2)gR - (m1 + m2)RẍR, which can be rearranged as ẍ = (m1 - m2)g. Thus, ẍ represents the ________ of the system.

acceleration Signup and view all the answers

Newton's second law can be used to show that ______

I O ω̇ = M O Signup and view all the answers

The mass moment of inertia I about a speciﬁed reference axis is deﬁned as ______

I = r 2 dm Signup and view all the answers

If the rotation axis of a homogeneous rigid body does not coincide with the body’s axis of symmetry, but is parallel to it at a distance d, then the mass moment of inertia about the rotation axis is given by the parallel-axis theorem, ______

I = Is + md 2 Signup and view all the answers

Let F be the tension in the cable. Because the drum rotation axis is assumed to be ﬁxed, we can use (3.2.1). Summing moments about the drum center gives ______.

M O = 300 − 0.2F Signup and view all the answers

Equivalent mass and inertia are complementary concepts. A system should be viewed as an equivalent if an external force is applied, and as an equivalent if an external torque is applied.

Signup and view all the answers

Equation (3.1.12) is a statement of conservation of _________ energy.

mechanical Signup and view all the answers

The dry friction force F is directly proportional to the force N normal to the frictional surface. Thus F = _______N.

$\mu$ Signup and view all the answers

For the mass m shown in Figure 3.1.3b, the normal force N must equal mg cos φ as long as the mass is in contact with the surface. The free body diagrams are shown in the ﬁgure for the two cases: v > 0 and v < 0. Newton’s second law applied in the direction parallel to the surface gives, for v > 0, m v̇ = f 1 − mg sin φ − _______mg cos φ v>0 (3)

$\mu$ Signup and view all the answers

Because the velocity is initially positive [v(0) = 3], we use equation (3) of Example 3.1.2: 2v̇ = f 1 − (sin 30◦ + 0.5 cos 30◦ )(2)(9.81) = f 1 − 18.3 For part (a), f 1 = 50 and thus v̇ = (50 − 18.3)/2 = 15.85 and the acceleration is positive. Thus, because v(0) > 0, the speed is always positive for t ≥ 0 and the mass never comes to rest. For part (b), f 1 = 5, v̇ = (5 − 18.3)/2 = −6.65, and thus the mass is decelerating. Because v(t) = −6.65t + 3, the speed becomes zero at t = 3/6.65 = 0.45 s.

$\mu$ Signup and view all the answers

Equivalent mass and inertia are complementary concepts. A system should be viewed as an equivalent mass if an external is applied, and as an equivalent inertia if an external is applied.

force, torque Signup and view all the answers

A pair of spur gears is shown in Figure 3.3.1. The input shaft (shaft 1) is connected to a motor that produces a torque T1 at a speed ω1 , and drives the output shaft (shaft 2). One use of such a system is to increase the effective motor . The gear ratio N is deﬁned as the ratio of the .

torque, output speed Signup and view all the answers

Several types of gears are used in mechanical drives. These include helical, spur, rack-and-pinion, worm, bevel, and ______ gears. Other mechanical drives use belts or chains.

planetary Signup and view all the answers

According to the linear equation (1), the characteristic roots are s = \pm j \sqrt{\frac{g}{L}} = \pm j \omega_n, where \omega_n = \sqrt{\frac{g}{L}}. The solution is \theta(t) = \theta(0) \cos(\omega_n t) + \sqrt{\frac{g}{L}}

frequency Signup and view all the answers

The pendulum will swing back and forth with a ______ frequency of \omega_n.

radian Signup and view all the answers

The work ______ by a moment M causing a rotation through an angle \theta is W = M d\theta.

done Signup and view all the answers

The tension ______ in the cable are approximately equal if I \thetä is negligible.

forces Signup and view all the answers

Study Notes

Electric Circuits

A mathematical description of the relation between the number of electrons (charge Q) and current i: i = Q/t
Unit of charge: Coulomb (C)
Unit of current: Ampere (A)
Unit of voltage: Volt (V)
Formula for power dissipated in a linear resistor: P = IV
Kirchhoff's current law: The sum of currents entering a node is equal to the sum of currents leaving a node
Kirchhoff's voltage law: The sum of voltage changes around a closed loop is zero
Series resistance law: R_total = R1 + R2 + ...
Model for capacitor voltage v1 in a series RC circuit: dv1/dt = (1/RC)v1
Expression for capacitor voltage v1(t) in a series RC circuit with a rectangular pulse input: v1(t) = V0(1 - e^(-t/RC))
Initial condition for capacitor voltage v1(0) in a series RC circuit with a rectangular pulse input: v1(0) = V0

More Electric Circuits

Expression for current in a circuit: i = ΔQ/Δt
Power in a circuit: P = VI
Expression for v3 as a function of v1 and v2: v3 = v1 + v2
Voltage vo as a function of applied voltage vs: vo = vs*R2/(R1+R2)
Current-divider rule: i1 = i_total * R2 / (R1 + R2)
Parallel resistance law for two resistors: 1/R_total = 1/R1 + 1/R2
Equation for voltage-current relation of a diode: i = IS(e^(v_D/V_T) - 1)
Energy stored in a capacitor: E = (1/2)CV^2
Impedance: Z = R + jX
Principles of direct-current (dc) motors
Power consumed by the load: P = v*i
Maximum value of the ratio r: r = v_o / v_i

Fluid Mechanics

Equation for mass flow rate: ṁ = ρAv
Unit of mass flow rate in FPS units: lbm/s
Unit of mass flow rate in SI units: kg/s
Equation for hydrostatic pressure at the bottom of a column of fluid: p = ρgh
Model for motion of the displacement of the mass: ẍ = (m1 - m2)g
Expression for velocity of the displacement as a function of time: v = v0 + at
Expression for mass flow rate that must be delivered or absorbed by the two pumps
Expression for equivalent mass of the rack, pinion, and load
Difference between hydraulics and pneumatics
Conservation of mass principle: Δm = 0
Appropriate resistance relation for convection heat transfer: R = 1/hA
Linearized model corresponding to the relation (7.3.7)
Relation (7.3.8) in terms of p_r: p_r = p_1 - p_2
Combination rules for series and parallel fluid resistances
Equation for fluid mass stored in the container: Δm = ρΔV
Alternative hydraulic models of a container of fluid
Equation for capacitance of a V-shaped trough: C = ΔV / Δp
Linearized resistance in terms of pressure and mass flow rate: R = Δp / ṁ
Expression for force f_3 in terms of force f_1: f_3 = f_1

Mechanics

Newton's first law: a particle originally at rest, or moving in a straight line with a constant speed, will remain that way as long as it is not acted upon by an unbalanced external force
Newton's third law: every action is opposed by an equal and opposite reaction
Newton's second law: F = ma
Conservation of mechanical energy: ΔE = 0
Term on the left-hand side of the equal sign: kinetic energy (KE)
If the work done by f(x) is independent of the path and depends only on the end points, then the force f(x) is derivable from a function V(x)
According to Newton's law for mass m1, the equation of motion is given by m1ẍ = m1g - T1
The equation of motion for mass m2 is m2ÿ = T2 - m2g
The equation of motion for the pulley system is ẍI = (m1 - m2)gR - (m1 + m2)RẍR
Mass moment of inertia I about a specified reference axis is defined as I = ∫r^2 dm
If the rotation axis of a homogeneous rigid body does not coincide with the body's axis of symmetry, but is parallel to it at a distance d, then the mass moment of inertia about the rotation axis is given by the parallel-axis theorem: I = I_cm + md^2
Dry friction force F is directly proportional to the force N normal to the frictional surface: F = μN
Equation (3.1.12) is a statement of conservation of mechanical energy: ΔE = 0
Equivalent mass and inertia are complementary concepts
A system should be viewed as an equivalent mass if an external force is applied, and as an equivalent inertia if an external torque is applied
A pair of spur gears is shown in Figure 3.3.1. The input shaft (shaft 1) is connected to a motor that produces a torque T1 at a speed ω1, and drives the output shaft (shaft 2).

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Ch 6-7 Systems

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Questions and Answers

What is the mathematical description of the relation between the number of electrons (charge Q) and current i?

What is the unit of charge?

What is the unit of current?

What is the unit of voltage?

What is the formula for power dissipated in a linear resistor?

What is Kirchhoff's current law?

What is Kirchhoff's voltage law?

What is the series resistance law?

What is the model for the capacitor voltage v1 in the series RC circuit?

What is the expression for the capacitor voltage v1 (t) in the series RC circuit with a rectangular pulse input?

What is the initial condition for the capacitor voltage v1 (0) in the series RC circuit with a rectangular pulse input?

What is the current in the circuit?

What is the power in the circuit?

Derive the expression for $v3$ as a function of $v1$ and $v2$.

Obtain the voltage $vo$ as a function of the applied voltage $vs$.

What is the equation for the current-divider rule?

What is the equation for the parallel resistance law for two resistors?

What is the equation for the voltage-current relation of a diode?

What is the equation for the energy stored in a capacitor?

What are Kirchhoff's current and voltage laws?

What is impedance and how is it related to electrical resistance?

What are the principles of direct-current (dc) motors?

What is the power consumed by the load in terms of vS?

What is the maximum value of the ratio r?

What is the input-output relation for vo as a function of vi?

What are the solutions for the currents i1, i2, and i3 in terms of the given voltages and resistances?

What is the equation for mass flow rate?

What is the unit of mass flow rate in FPS units?

What is the unit of mass flow rate in SI units?

What is the equation for hydrostatic pressure at the bottom of a column of fluid?

What is the model for the motion of the displacement of the mass in part (a) of the figure?

What is the expression for the velocity of the displacement as a function of time?

What is the expression for the mass flow rate that must be delivered or absorbed by the two pumps?

What is the expression for the equivalent mass of the rack, pinion, and load?

What is the difference between hydraulics and pneumatics?

What is the conservation of mass principle?

What is the appropriate resistance relation for convection heat transfer?

What is the linearized model corresponding to the relation (7.3.7)?

How can the relation (7.3.8) be expressed in terms of $p_r$?

What are the combination rules for series and parallel fluid resistances?

What is the equation for the fluid mass stored in the container?

What are the alternative hydraulic models of a container of fluid?

What is the equation for the capacitance of a V-shaped trough?

What is the linearized resistance in terms of pressure and mass flow rate?

Obtain the expression for the force $f_3$ in terms of the force $f_1$.

What is the relation between the pressures $p_1$ and $p_2$ in the hydraulic brake system?

Explain the tradeoff for force amplification in the hydraulic brake system.

What is the equation for conservation of mass in terms of mass inflow rate and mass outflow rate?

What is the analogy between fluid mass accumulation and electrical capacitance?

What is the analogy between Kirchhoff's voltage law and the compatibility law in fluid systems?

What is the relation between fluid capacitance and the change in pressure?

How can the expression for the capacitance of a storage tank be derived?

Newton’s first law states that a particle originally at rest, or moving in a straight line with a constant speed, will remain that way as long as it is not acted upon by an unbalanced external ______.

The third law is summarized by the commonly used statement that every action is opposed by an equal ______.

For an object treated as a particle of mass m, the second law can be expressed as dv = ______

Conservation of mechanical energy is a direct consequence of ______'s second law.

The term on the left-hand side of the equal sign is called the ______ energy (KE).

If the work done by f (x) is independent of the path and depends only on the end points, then the force f (x) is derivable from a function V (x) as follows: dV (3.1.5) f (x) = − dx Then, in this case, f (x) is called a ______ force.

This equation shows that V (x) has the same units as ______ energy.

According to Newton's law for mass m1, the equation of motion is given by m1ẍ = m1g - T1, where T1 is the ________ force in the cable.

The equation of motion for mass m2 is m2ÿ = T2 - m2g. T2 is the ________ force in the cable.

The equation of motion for mass m2 can be expressed in terms of the inertia I as Iθ̈ = RT1 - RT2 = R(T1 - T2). This equation represents the ________ law for mass m2.

The equation of motion for the pulley system is ẍI = (m1 - m2)gR - (m1 + m2)RẍR, which can be rearranged as ẍ = (m1 - m2)g. Thus, ẍ represents the ________ of the system.

Newton's second law can be used to show that ______

The mass moment of inertia I about a speciﬁed reference axis is deﬁned as ______

If the rotation axis of a homogeneous rigid body does not coincide with the body’s axis of symmetry, but is parallel to it at a distance d, then the mass moment of inertia about the rotation axis is given by the parallel-axis theorem, ______

Let F be the tension in the cable. Because the drum rotation axis is assumed to be ﬁxed, we can use (3.2.1). Summing moments about the drum center gives ______.

Equivalent mass and inertia are complementary concepts. A system should be viewed as an equivalent ______ if an external force is applied, and as an equivalent ______ if an external torque is applied.

Equation (3.1.12) is a statement of conservation of _________ energy.

The dry friction force F is directly proportional to the force N normal to the frictional surface. Thus F = _______N.

Equivalent mass and inertia are complementary concepts. A system should be viewed as an equivalent mass if an external ______ is applied, and as an equivalent inertia if an external ______ is applied.

Several types of gears are used in mechanical drives. These include helical, spur, rack-and-pinion, worm, bevel, and ______ gears. Other mechanical drives use belts or chains.

According to the linear equation (1), the characteristic roots are s = \pm j \sqrt{\frac{g}{L}} = \pm j \omega_n, where \omega_n = \sqrt{\frac{g}{L}}. The solution is \theta(t) = \theta(0) \cos(\omega_n t) + \sqrt{\frac{g}{L}}

The pendulum will swing back and forth with a ______ frequency of \omega_n.

The work ______ by a moment M causing a rotation through an angle \theta is W = M d\theta.

The tension ______ in the cable are approximately equal if I \thetä is negligible.

Equivalent mass and inertia are complementary concepts. A system should be viewed as an equivalent if an external force is applied, and as an equivalent if an external torque is applied.

Equivalent mass and inertia are complementary concepts. A system should be viewed as an equivalent mass if an external is applied, and as an equivalent inertia if an external is applied.