1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
love history [14]
3 years ago
12

(a) According to Hooke's Law, the force required to hold any spring stretched x meters beyond its natural length is f(x)=kx. Sup

pose a spring has a natural length of 20 cm. If a 25-N force is required to keep it stretched to a length of 30 cm, how much work is required to stretch it from 20 cm to 25 cm?
(b) Find the area of the region enclosed by one loop of the curve r=2sin(5θ).
Physics
1 answer:
KengaRu [80]3 years ago
5 0

Answer:

a) The work required to stretch the spring from 20 centimeters to 25 centimeters is 0.313 joules, b) The area of the region enclosed by one loop of the curve r(\theta) = 2\cdot \sin 5\theta is 4\pi.

Explanation:

a) The work, measured in joules, is a physical variable represented by the following integral:

W = \int\limits^{x_{f}}_{x_{o}} {F(x)} \, dx

Where

x_{o}, x_{f} - Initial and final position, respectively, measured in meters.

F(x) - Force as a function of position, measured in newtons.

Given that F = k\cdot x and the fact that F = 25\,N when x = 0.3\,m - 0.2\,m, the spring constant (k), measured in newtons per meter, is:

k = \frac{F}{x}

k = \frac{25\,N}{0.3\,m-0.2\,m}

k = 250\,\frac{N}{m}

Now, the work function is obtained:

W = \left(250\,\frac{N}{m} \right)\int\limits^{0.05\,m}_{0\,m} {x} \, dx

W = \frac{1}{2}\cdot \left(250\,\frac{N}{m} \right)\cdot [(0.05\,m)^{2}-(0.00\,m)^{2}]

W = 0.313\,J

The work required to stretch the spring from 20 centimeters to 25 centimeters is 0.313 joules.

b) Let be r(\theta) = 2\cdot \sin 5\theta. The area of the region enclosed by one loop of the curve is given by the following integral:

A = \int\limits^{2\pi}_0 {[r(\theta)]^{2}} \, d\theta

A = 4\int\limits^{2\pi}_{0} {\sin^{2}5\theta} \, d\theta

By using trigonometrical identities, the integral is further simplified:

A = 4\int\limits^{2\pi}_{0} {\frac{1-\cos 10\theta}{2} } \, d\theta

A = 2 \int\limits^{2\pi}_{0} {(1-\cos 10\theta)} \, d\theta

A = 2\int\limits^{2\pi}_{0}\, d\theta - 2\int\limits^{2\pi}_{0} {\cos10\theta} \, d\theta

A = 2\cdot (2\pi - 0) - \frac{1}{5}\cdot (\sin 20\pi-\sin 0)

A = 4\pi

The area of the region enclosed by one loop of the curve r(\theta) = 2\cdot \sin 5\theta is 4\pi.

You might be interested in
A scene in a movie has a stuntman falling through a floor onto a bed in the room below. The plan is to have the actor fall on hi
tekilochka [14]

Answer:

The maximum mass that can fall on the mattress without exceeding the maximum compression distance is 16.6 kg

Explanation:

Hi there!

Due to conservation of energy, the potential energy (PE) of the mass at a height of 3.32 m will be transformed into elastic potential energy (EPE) when it falls on the mattress:

PE = EPE

m · g · h = 1/2 k · x²

Where:

m = mass.

g = acceleration due to gravity.

h = height.

k = spring constant.

x = compression distance

The maximum compression distance is 0.1289 m, then, the maximum elastic potential energy will be the following:

EPE =1/2 k · x²

EPE = 1/2 · 65144 N/m · (0.1289 m)² = 541.2 J

Then, using the equation of gravitational potential energy:

PE = m · g · h =  541.2 J

m =  541.2 J/ g · h

m = 541.2 kg · m²/s² / (9.8 m/s² · 3.32 m)

m = 16.6 kg

The maximum mass that can fall on the mattress without exceeding the maximum compression distance is 16.6 kg.

6 0
3 years ago
How to atoms behave in non-magnetic items?
Anastaziya [24]

Answer:

By altering the quantum interactions of the electrons in the atoms of a metal's atoms, scientists from the University of Leeds have generated magnetism in metals that aren’t normally magnetic.

Explanation:

5 0
3 years ago
Calculate, using a diagram, the resultant of the following vector combinations:
Veronika [31]
Add them together with south being negative. (-350 + 25) to get 325 south
4 0
3 years ago
Please help !!!!!!!!!!!!’
Zanzabum

Answer:

GYU

Explanation:UGGGGGGGGGIBKLJ

6 0
3 years ago
Suppose there are 100,000 atoms of a radioactive substance that has a ½ life of 10 minutes. How many atoms will remain after 40
dezoksy [38]

Answer:

c. 12,500

Explanation:

Original number of atoms = 100,000 atoms

Half- life  = 10min

Unknown:

The number of atoms that will remain after 10min  = ?

Solution:

The half - life is the time taken for half of a radioactive substance to decay by half.

    Time taken        Number of atom   half life

           10min             100000                   _

           20min             50000                    1

           30min             25000                    2

           40min              12500                     3

6 0
2 years ago
Other questions:
  • A student uses an electronic force sensor to study how much force the student’s finger can apply to a specific location. The stu
    12·2 answers
  • Una cuerda de violin vibra con una frecuencia fundamental de 435 Hz. Cual sera su frecuencia de vibracion si se le somete a una
    15·1 answer
  • If the pressure inside a balloon was 15 Pa and the volume was 120 ml, what is the volume if the pressure increases to 400 ml? Pl
    15·2 answers
  • When more than one wave is in the same location at the same time, then there is _____ between the waves?
    13·1 answer
  • A ball dropped from rest falls freely until it hits the ground with a speed pf 20 m/s. The time during with the ball is in free
    11·1 answer
  • During bicycling, a 70 kg person's body produces energy at a rate of about 500 W due to metabolism, 80% of which is converted to
    14·1 answer
  • A plane passes over Point A with a velocity of 8000 m/s north. Forty seconds later it passes over Point B at a velocity of 10,00
    10·1 answer
  • En una sala de juntas hay mesas, sillas y otras personas. ¿Cuál de ellas tienen temperaturas
    8·1 answer
  • A skateboarder rolls horizontally off the top of a staircase and lands at the bottom. The staircase has a horizontal length of 1
    11·1 answer
  • The speed that a tsunami can travel is modeled by the equation , where s is the speed in kilometers per hour and d is the averag
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!