You have a series circuit powered by a 9V battery. If you double the amount of

A spring attached to the ceiling is stretched one foot by a four pound weight. The mass is set in motion by pulling down 2 feet

Marina CMI [18]

Given that,

Weight = 4 pound

$W=4\ lb$

Stretch = 2 feet

Let the force be F.

The elongation of the spring after the mass attached is

$x=2-1=1\ feet$

(a). We need to calculate the value of spring constant

Using Hooke's law

$F=kx$

$k=\dfrac{F}{x}$

Where, F = force

k = spring constant

x = elongation

Put the value into the formula

$k=\dfrac{4}{1}$

$k=4$

(b). We need to calculate the mass

Using the formula

$F=mg$

$m=\dfrac{F}{g}$

Where, F = force

g = acceleration due to gravity

Put the value into the formula

$m=\dfrac{4}{32}$

$m=\dfrac{1}{8}\ lb$

We need to calculate the natural frequency

Using formula of natural frequency

$\omega=\sqrt{\dfrac{k}{m}}$

Where, k = spring constant

m = mass

Put the value into the formula

$\omega=\sqrt{\dfrac{4}{\dfrac{1}{8}}}$

$\omega=\sqrt{32}$

$\omega=4\sqrt{2}$

(c). We need to write the differential equation

Using differential equation

$m\dfrac{d^2x}{dt^2}+kx=0$

Put the value in the equation

$\dfrac{1}{8}\dfrac{d^2x}{dt^2}+4x=0$

$\dfrac{d^2x}{dt^2}+32x=0$

(d). We need to find the solution for the position

Using auxiliary equation

$m^2+32=0$

$m=\pm i\sqrt{32}$

We know that,

The general equation is

$x(t)=A\cos(\sqrt{32t})+B\sin(\sqrt{32t})$

Using initial conditions

(I). $x(0)=2$

Then, $x(0)=A\cos(\sqrt{32\times0})+B\sin(\sqrt{32\times0})$

Put the value in equation

$2=A+0$

$A=2$ .....(I)

Now, on differentiating of general equation

$x'(t)=-\sqrt{32}A\sin(\sqrt{32t})+\sqrt{32}B\cos(\sqrt{32t})$

Using condition

(II). $x'(0)=0$

Then, $x'(0)=-\sqrt{32}A\sin(\sqrt{32\times0})+\sqrt{32}B\cos(\sqrt{32\times0})$

Put the value in the equation

$0=0+\sqrt{32}B$

So, B = 0

Now, put the value in general equation from equation (I) and (II)

So, The general solution is

$x(t)=2\cos\sqrt{32t}$

(e). We need to calculate the time

Using formula of time

$T=\dfrac{2\pi}{\omega}$

Put the value into the formula

$T=\dfrac{2\pi}{4\sqrt{2}}$

$T=1.11\ sec$

Hence, (a). The value of spring constant is 4.

(b). The natural frequency is 4√2.

(c). The differential equation is $\dfrac{d^2x}{dt^2}+32x=0$

(d). The solution for the position is $x(t)=2\cos\sqrt{32t}$

(e). The time period is 1.11 sec.

5 0

3 years ago

When magma is cooled, it changes into which of the following?

vekshin1

Igneous rock

Explanation:

4 0

3 years ago

Read 2 more answers

Define 1 unit electricity

Tomtit [17]

Answer:

A unit is represented in kWH or Kilowatt Hour. This is the actual electricity or energy used. If you use 1000 Watts or 1 Kilowatt of power for 1 hour then you consume 1 unit or 1 Kilowatt-Hour (kWh) of electricity.

4 0

3 years ago

Two resistors of 12Ω and 2Ω are connected in parallel to a battery of 4 V. Current through 12Ω resistor will be _____________ an

Alex

Answer:

0.33 Amp

8 Watts

Explanation:

The current through the 12 Ω resistor will be given by Ohm's law:

4 V / 12 Ω = 1/3 Amp ≈ 0.33 Amp

The power dissipated on the 2 Ω resistor can be calculated via the formula: $\frac{V^2}{R}$ which gives:

4^2 / 2 = 8 Watts

6 0

3 years ago

After rubbing a balloon on your shirt, your hair sticks up when the balloon is near your head. Explain how tiny particles on you

skad [1K]

Almost right. protons are positive and electrons are negative. so when you run the balloon on your hair, electrons are transferred between them (i’m not sure which direction) and now one is positively charged as it lost negative particles and one is negative as it gained negative particles (electrons). opposite charges attract

3 0

2 years ago