(a) 1200 rad/s
The angular acceleration of the rotor is given by:
[tex]\alpha = \frac{\omega_f - \omega_i}{t}[/tex]
where we have
[tex]\alpha = -80.0 rad/s^2[/tex] is the angular acceleration (negative since the rotor is slowing down)
[tex]\omega_f [/tex] is the final angular speed
[tex]\omega_i = 2000 rad/s[/tex] is the initial angular speed
t = 10.0 s is the time interval
Solving for [tex]\omega_f[/tex], we find the final angular speed after 10.0 s:
[tex]\omega_f = \omega_i + \alpha t = 2000 rad/s + (-80.0 rad/s^2)(10.0 s)=1200 rad/s[/tex]
(b) 25 s
We can calculate the time needed for the rotor to come to rest, by using again the same formula:
[tex]\alpha = \frac{\omega_f - \omega_i}{t}[/tex]
If we re-arrange it for t, we get:
[tex]t = \frac{\omega_f - \omega_i}{\alpha}[/tex]
where here we have
[tex]\omega_i = 2000 rad/s[/tex] is the initial angular speed
[tex]\omega_f=0[/tex] is the final angular speed
[tex]\alpha = -80.0 rad/s^2[/tex] is the angular acceleration
Solving the equation,
[tex]t=\frac{0-2000 rad/s}{-80.0 rad/s^2}=25 s[/tex]
Two square air-filled parallel plates that are initially uncharged are separated by 1.2 mm, and each of them has an area of 190 mm^2. How much charge must be transferred from one plate to the other if 1.1 nJ of energy are to be stored in the plates? ( ε0 = 8.85 × 10^-12 C2/N · m^2)
Answer:
[tex]5.5\cdot 10^{-11} C[/tex]
Explanation:
The capacitance of the parallel-plate capacitor is given by:
[tex]C=\epsilon_0 \frac{A}{d}[/tex]
where
[tex]\epsilon_0 = 8.85\cdot 10^{-12} F/m[/tex] is the vacuum permittivity
[tex]A=190 mm^2 = 190 \cdot 10^{-6} m^2[/tex] is the area of the plates
[tex]d=1.2 mm = 0.0012 m[/tex] is the separation between the plates
Substituting,
[tex]C=(8.85\cdot 10^{-12}) \frac{190 \cdot 10^{-6}}{0.0012}=1.4\cdot 10^{-12}F[/tex]
The energy stored in the capacitor is given by
[tex]U=\frac{Q^2}{2C}[/tex]
Since we know the energy
[tex]U=1.1 nJ = 1.1 \cdot 10^{-9} J[/tex]
we can re-arrange the formula to find the charge, Q:
[tex]Q=\sqrt{2UC}=\sqrt{2(1.1\cdot 10^{-9} J)(1.4\cdot 10^{-12}F )}=5.5\cdot 10^{-11} C[/tex]
The charge needed to be transferred between the plates to store 1.1 nJ of energy is approximately 55.5 pC.
To determine the charge that must be transferred between two initially uncharged parallel plates to store 1.1 nJ of energy, we use the concept of a parallel plate capacitor. The energy stored in a capacitor is given by the formula:
U = 0.5 * C * V²
Where U is the energy (1.1 nJ = 1.1 × 10⁻⁹J), C is the capacitance, and V is the potential difference between the plates.
Step 1: Calculate the Capacitance
Capacitance (C) for a parallel-plate capacitor is given by:
C = (0 * A) / d
Where 0 is the permittivity of free space (8.85 × 10⁻¹² C²/N·m²), A is the area of the plates (190 mm² = 190 × 10⁻⁶ m²), and d is the separation between the plates (1.2 mm = 1.2 × 10⁻³ m).
Substitute the values:
C = (8.85 × 10⁻¹²* 190 × 10⁻⁶) / 1.2 × 10⁻³
C ≈ 1.4 × 10⁻¹² F
Step 2: Calculate the Voltage
Using the energy formula, solve for V:
1.1 × 10⁻⁹ J = 0.5 * 1.4 × 10⁻¹² F * V²
V² = (1.1 × 10⁻⁹ J) / (0.5 * 1.4 × 10⁻¹²F)
V² ≈ 1571 J/F
V ≈ 39.65 V
Step 3: Calculate the Charge
Use the relationship Q = C * V:
Q = 1.4 × 10⁻¹² F * 39.65 V
Q ≈ 5.55 × 10⁻¹¹ C
Therefore, approximately 55.5 pC (picoCoulombs) of charge must be transferred from one plate to the other.
A Ferris wheel has diameter of 10 m and makes one revolution in 8.0 seconds. A woman weighing 670 N is sitting on one of the benches attached at the rim of the wheel. What is the net force on this woman as she passes through the highest point of her motion?
Answer:
208 N
Explanation:
The net force on the woman is equal to the centripetal force, which is given by
[tex]F=m\frac{v^2}{r}[/tex]
where
m is the mass of the woman
v is her speed
r is the radius of the wheel
here we have:
r = d/2 = 5 m is the radius of the wheel
[tex]m=\frac{W}{g}=\frac{670 N}{9.8 m/s^2}=68.4 kg[/tex] is the mass of the woman (equal to her weight divided by the acceleration of gravity)
The wheel makes one revolution in t=8.0 s, so the speed is:
[tex]v=\frac{2\pi r}{t}=\frac{2\pi (5.0 m)}{8.0 s}=3.9 m/s[/tex]
so now we can find the centripetal force:
[tex]F=(68.4 kg)\frac{(3.9 m/s)^2}{5.0 m}=208 N[/tex]
A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes a time of 2.70 s for the boat to travel from its highest point to its lowest, a total distance of 0.700 m. The fisherman sees that the wave crests are spaced a horizontal distance of 6.50 m apart. a. How fast are the waves traveling? b. What is the amplitude A of each wave?
Answer: a. 1.203 m/s b.0.35m
Explanation:
a. Velocity of the waves
The velocity [tex]V[/tex] of a wave is given by the following equation:
[tex]V=\frac{\lambda}{T}[/tex] (1)
Where:
[tex]\lambda=6.50m[/tex] is the wavelength (the horizontal space between crest and crest)
[tex]T=2.70s.2=5.4s[/tex] is the period of the wave (If we were told it takes a time of 2.70 s for the boat to travel from its highest point to its lowest, the period is twice this amount, which is the time it takes to travel one wavelength)
Substituting the known values:
[tex]V=\frac{6.50m}{5.4s}=1.203m/s[/tex] (2)
[tex]V=1.203m/s[/tex] Velocity of the wave
b. Amplitude of each waveThe amplitude [tex]A[/tex] of a wave is defined is given by the following equation:
[tex]A=\frac{maximumpoint-minimumpoint}{2}[/tex] (3)
If we know the total distance between the highest point to the lowest point is 0.7 m. This means:
[tex]maximumpoint-minimumpoint=0.7m[/tex]
Substituting this value in (3):
[tex]A=\frac{0.7m}{2}[/tex] (4)
[tex]A=0.35m[/tex] This is the amplitude of the wave
The waves for the boat of fisherman is travelling with the speed of 1.203 m/s and the amplitude of each wave is 0.35 m.
What is speed of wave?Speed of wave is the rate of speed by which the wave travel the distance in the time taken by it.
The distance traveled by the boat from its highest point to the lowest point is 0.700 m. The time taken by the boat is 2.70.
The fisherman sees that the wave crests are spaced a horizontal distance of 6.50 m apart. As, the horizontal distance of crests is the wavelength of the wave. Thus, the wavelength of the wave is,
[tex]\lambda=6.50\rm s[/tex]
The period of the wave is twice the time taken from highest to lowest point. Thus, the time period of the wave is,
[tex]T=2\times2.7\\T=5.4\rm s[/tex]
The speed of the wave is the ratio of wavelength to the time period. Thus, the speed of the waves travelling is,
[tex]v=\dfrac{6.50}{5.4}\\v=1.203\rm m/s[/tex]
The amplitude of wave is the half of the difference of highest point to the lowest point of the wave.
Now, the distance traveled by the boat from its highest point to the lowest point is 0.700 m. Thus, the amplitude of each wave is,
[tex]A=\dfrac{0.7}{2}A=0.35\rm m[/tex]
Thus, the waves for the boat of fisherman is travelling with the speed of 1.203 m/s and the amplitude of each wave is 0.35 m.
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A resultant vector is 5 units long and makes an angle of 23 degrees measured counter-clockwise with respect to the positive x-axis. What are the magnitude and angle (measured counter-clockwise with respect to the positive x-axis) of the equilibrant vector?
Answer:
5 units at 203 degrees
Explanation:
The equilibrant vector must have components that are opposite to those of the initial vector.
The components of the initial vector are:
[tex]v_x = v cos \theta = 5 cos 23^{\circ}=4.60\\v_y = v sin \theta = 5 sin 23^{\circ} = 1.95[/tex]
So the components of the equilibrant vector must be
[tex]v_x = -4.60\\v_y = -1.95[/tex]
which means its magnitude is
[tex]v=\sqrt{v_x^2 + v_y^2}= 5[/tex] (same magnitude as the initial vector)
and it is located in the 3rd quadrant, so its angle will be
[tex]\theta = 180^{\circ} + tan^{-1} (\frac{1.95}{4.60})=203^{\circ}[/tex]
What is the emf through a single coil of wire if the magnetic flux changes from -57 Wb to +43 Wb in 0.17 s? O 588 V O 344 V O 386 V O 496 V
Answer:
Emf through a single coil of wire is 588 V.
Explanation:
We need to find the emf through a single coil of wire if the magnetic flux changes from -57 Wb to +43 Wb in 0.17 s
According to Faraday's law, emf of coil is directly proportional to the rate of change of magnetic flux. It can be written as :
[tex]\epsilon=\dfrac{d\phi}{dt}[/tex]
Initial flux, [tex]\phi_1=-57\ Wb[/tex]
Final flux, [tex]\phi_2=+43\ Wb[/tex]
So, [tex]\epsilon=\dfrac{\phi_2-\phi_1}{dt}[/tex]
[tex]\epsilon=\dfrac{43\ Wb-(-57\ Wb)}{0.17\ s}[/tex]
[tex]\epsilon=588.23\ V[/tex]
or
[tex]\epsilon=588\ V[/tex]
So, the EMF through a single coil of wire is 588 V. Hence, this is the required solution.
An electroplating solution is made up of nickel(II) sulfate. How much time would it take to deposit 0.500 g of metallic nickel on a custom car part using a current of 3.00 A
To calculate the time required to deposit 0.500 g of metallic nickel on a custom car part using a current of 3.00 A, you can use the equation Time (s) = Mass (g) / (Current (A) × Charge (C/g)). Calculations involving the charge of nickel and the given values can be used to determine the time in seconds. Converting the time to minutes or hours is also possible.
Explanation:To calculate the time required to deposit 0.500 g of metallic nickel using a current of 3.00 A, we need to use the equation:
Time (s) = Mass (g) / (Current (A) × Charge (C/g))
For nickel, the charge is 2+ and the atomic mass is approximately 58.69 g/mol. From these values, we can calculate the charge in C/g:
Charge (C/g) = (2 × 1.602 × 10-19 C) / (58.69 g/mol)
Using this value and the given mass and current, we can calculate the time required in seconds:
Time (s) = 0.500 g / (3.00 A × Charge (C/g))
Converting the time to minutes or hours can be done by dividing by 60 or 3600, respectively.
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Write an equation that expresses the following relationship. u varies jointly with p and d and inversely with w In your equation, use k as the constant of proportionality.
[tex]\boxed{y=k\frac{pd}{w}}[/tex]
Explanation:Let's explain what direct and indirect variation mean:
When we say that [tex]y[/tex] varies jointly as [tex]x \ and \ w[/tex], we mean that:[tex]y=kxw[/tex] for some nonzero constant [tex]k[/tex] that is the constant of variation or the constant of proportionality.
On the other hand, when we say that [tex]y[/tex] varies inversely as [tex]x[/tex] or [tex]y[/tex] is inversely proportional to [tex]x[/tex], we mean that:[tex]y=\frac{k}{x}[/tex] for some nonzero constant [tex]k[/tex], where [tex]k[/tex] is also the constant of variation.
___________________
In this problem, [tex]u[/tex] varies jointly with [tex]p[/tex] and [tex]d[/tex] and inversely with [tex]w[/tex], being [tex]k[/tex] the constant of proportionality, then:
[tex]\boxed{y=k\frac{pd}{w}}[/tex]
The equation expressing the described relationship is u = kpd/w. This represented u varying jointly with p and d and inversely with w, with k being the constant of proportionality.
Explanation:The equation that expresses the relationship where 'u' varies jointly with 'p' and 'd' and inversely with 'w' using 'k' as the constant of proportionality would be: u = kpd/w. Here, 'k' is the constant of proportionality which determines the rate at which 'u' varies relative to 'p', 'd', and 'w'. Joint variability is expressed by multiplying the variables 'p' and 'd', while inverse proportionality is shown through dividing by 'w'.
This equation effectively expresses the described relationship - as 'p' or 'd' increase (or 'w' decrease), 'u' increases and vice versa. It’s important to understand that the constant 'k' would need to be determined based on additional context or data.
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