Answer: no, it is an opinion and is not testable (a)
Explanation:
i just got this question on Study Island
In this section of a circuit, a current of 2.6 A flows across R1. What is the potential difference V between point x and point y (across R2)? Let R1 = 4.0 ohm, R2 = 8.0 ohm, and R3 = 1.0 ohm.
Answer:
V=21V
Explanation:
In series combination current is same
I=2.6A
R2=8ohm
V2=2.6×8
V2=20.8
V2=21V
Answer:
Explanation:
21 v
The total amount of energy contained in an object.
Answer: see the explanation
Explanation:
The total amount of energy contained in an object is called its energy. Energy can exist in different forms, such as kinetic energy (energy of motion), potential energy (stored energy), thermal energy (heat), chemical energy (energy stored in chemical bonds), and many others.
To calculate the total energy of an object, you need to consider the different forms of energy it possesses. For example, if an object is moving, it has kinetic energy, which depends on its mass and velocity. The equation to calculate kinetic energy is KE = 0.5 * mass * velocity^2, where KE represents kinetic energy, mass is the object's mass, and velocity is its speed.
If an object is at a certain height above the ground, it has gravitational potential energy. The equation for gravitational potential energy is PE = mass * gravity * height, where PE represents potential energy, mass is the object's mass, gravity is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth), and height is the object's height above a reference point.
Additionally, an object can have other forms of energy, such as thermal energy due to its temperature, chemical energy stored in its chemical bonds, or electrical energy if it has an electric charge.
It's important to note that the total energy of an object is the sum of all the different forms of energy it possesses. Each form of energy can be converted into another form, but the total energy remains constant, as stated by the law of conservation of energy.
In summary, the total amount of energy contained in an object is the sum of its different forms of energy, such as kinetic energy, potential energy, thermal energy, chemical energy, and others. To calculate the total energy, you need to consider the specific forms of energy present and use the appropriate equations. The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another.
can u tell in a very easy where a small child can also understand
Brainly Ginny character avatar
AI-generated answer
The total amount of energy in an object is all the different kinds of energy that it has. Energy is what makes things happen and allows them to do work. It can come in different forms, like when you're running and have energy to move, or when you have a ball that can bounce and roll.
Imagine you have a toy car that you wind up. When you wind it up, you're putting energy into it. That energy gets stored inside the car. When you let go of the car, it starts moving and uses up that stored energy. This is an example of kinetic energy, which is the energy of motion.
Another kind of energy is potential energy. Let's say you have a ball on top of a hill. The ball has potential energy because it can roll down the hill. When you let the ball go, it starts rolling and its potential energy turns into kinetic energy as it moves.
Objects can also have thermal energy, which is heat energy. When you touch something hot, like a stove, you feel the thermal energy transferring to your hand.
Some objects have chemical energy stored inside them. For example, when you eat food, your body breaks it down and turns it into energy that you can use.
These are just a few examples of the different forms of energy. The total energy of an object is the sum of all the different kinds of energy it has. Energy can change from one form to another, but it cannot be created or destroyed.
So, to put it simply, the total energy of an object is all the different types of energy it has, like the energy of motion, heat, stored energy, and more
Juliana walks 45 meters East, 45 meters south and 45 meters north. What is her resultant displacement?
Her resultant displacement is (45 Meters East.)
She originally walks 45 meters east, then she goes south 45 meters, then north 45 meters. The south and north placements just make her go back to where her previous placements were before them. Making her 45 meters east.
Identify the area where convection is taking place.
In this area the convection is taking place inside the water which is inside the utensils.
Explanation:
Convection is the type of heat transfer method which is usually taking place through the liquid and gases. In this convection process, the liquid acts as the medium for the heat transfer. From the given picture we can see both convection and conduction. The conduction process which occurs in the solid objects. Here the heat is directly given to the pan which is a solid conductor so the pan conducts the heat. Inside the pan there happens to be the liquid which favors the convection process and again the eggs are heated by the conduction process as the outer part of the egg is solid.Work is the product of ___
_ and an object's displacement.
Answer:
Force
Explanation:
W= F * d
Work is defined in physics as the product of force and an object's displacement. It is expressed mathematically as Work = Force x Distance. The direction of the force and displacement also affect whether the work done is considered positive or negative.
Explanation:In physics, work is defined as the product of force and an object's displacement. It is represented mathematically as Work = Force x Distance. In this equation, Force is the amount of effort exerted on an object, and Distance or displacement is the amount of space over which the force is applied. If the force and the displacement are in the same direction, the work done is positive.
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Can someone pleaseeee answer this !!!!!!
Answer:
The person with locked legs will experience greater impact force.
Explanation:
Let the two persons be of nearly equal mass (say m)
The final velocity of an object (person) dropped from a height H (here 2 meters) is given by,
[tex]v=\sqrt{2gH}[/tex]
([tex]g[/tex] = acceleration due to gravity)
which can be derived from Newton's equation of motion,
[tex]v^2=u^2+2aS[/tex]
Now, the time taken (say [tex]t[/tex] ) for the momentum ( [tex]mv[/tex] ) to change to zero will be more in the case of the person who bends his legs on impact than who keeps his legs locked.
We know that,
[tex]Force=\frac{\Delta(mv)}{t}[/tex]
Naturally, the person who bends his legs will experience lesser force since [tex]t[/tex] is larger.
why does a passenger jumping out of a rapidly moving bus fall forqard with his face downwards
Answer:
Explanation:
It's Newton's first law of motion, and object in motion continues to stay in motion unless acted upon by an unbalanced force. In this case, your body is the force that's moving on the bus. When you jump(the unbalanced force) your body is still used to being in motion, which makes you fall and not land on your feet.
Bone has a Young's modulus of about
1.8 x 100 Pa. Under compression, it can
withstand a stress of about 1.58 x 10° Pa be-
fore breaking.
Assume that a femur (thigh bone) is 0.54 m
long, and calculate the amount of compression
this bone can withstand before breaking.
Answer in units of mm.
Answer: 4.74 mm
Explanation:
We can solve this problem with the following equation:
[tex]Y=\frac{stress}{strain}[/tex] (1)
Where:
[tex]Y=1.8(10)^{10} Pa[/tex] is the Young modulus for femur
[tex]stress=\frac{F}{A}=1.58(10)^{8} Pa[/tex] is the stress (force [tex]F[/tex] applied per unit of transversal area [tex]A[/tex]) on the femur
[tex]strain=\frac{\Delta l}{l_{o}}[/tex]
Being:
[tex]\Delta l[/tex] the compression the femur can withstand before breaking
[tex]l_{o}=0.54 m[/tex] is the length of the femur without compression
Writing the data in equation (1):
[tex]Y=\frac{\frac{F}{A}}{\frac{\Delta l}{l_{o}}}[/tex] (2)
[tex]1.8(10)^{10} Pa=\frac{1.58(10)^{8} Pa}{\frac{\Delta l}{0.54 m}}[/tex] (3)
Isolating [tex]\Delta l[/tex]:
[tex]\Delta l=\frac{(1.58(10)^{8} Pa)(0.54 m)}{1.8(10)^{10} Pa}[/tex] (4)
[tex]\Delta l=0.00474 m[/tex] (5) This is the compression in meters
Converting this result to millimeters:
[tex]\Delta l=0.00474 m \frac{1000 mm}{1 m}=4.74 mm[/tex]
A 0.300 kg mass is attached to
a 26.6 N/m spring. It is pulled
0.120 m and released. What is
the speed of the mass when it is
0.0600 m from equilibrium?
(Unit = m/s)
To calculate the speed of the 0.300 kg mass when it is 0.0600 m from equilibrium on a 26.6 N/m spring, use the conservation of energy principle to find the kinetic energy at that point and then solve for velocity.
Explanation:The problem involves a 0.300 kg mass attached to a spring with a force constant of 26.6 N/m. The mass is pulled 0.120 m from equilibrium and released. To find the speed of the mass when it is 0.0600 m from equilibrium, we can use the principle of conservation of energy. The total mechanical energy is conserved because we are ignoring any external forces like friction or air resistance.
Initially, when the mass is released from 0.120 m, the spring has some potential energy (due to being stretched) and the kinetic energy is zero since it's released from rest. When the mass is at 0.0600 m from equilibrium, there is a combination of potential energy (due to being displaced from equilibrium) and kinetic energy (since the mass is moving).
The initial potential energy (PEi) stored in the spring can be calculated using the formula PEi = 1/2 * k * x2, where k is the spring constant and x is the initial displacement. The final potential energy (PEf) is calculated similarly, with x being the final displacement of 0.0600 m.
The difference between the initial and final potential energy will give us the kinetic energy at 0.0600 m from equilibrium, since energy is conserved and the difference must have been converted to kinetic energy. The kinetic energy (KE) can be expressed as KE = 1/2 * m * v2, where m is the mass and v is the velocity we want to find.
Here's how you would set up the equations:
Calculate the initial potential energy: PEi = 1/2 * 26.6 N/m * (0.120 m)2.Calculate the final potential energy: PEf = 1/2 * 26.6 N/m * (0.0600 m)2.Find the kinetic energy at 0.0600 m by subtracting the final potential energy from the initial potential energy.Finally, solve for v in the kinetic energy equation to get the velocity of the mass at 0.0600 m from equilibrium.The speed of a 0.300 kg mass attached to a 26.6 N/m spring, when it is 0.0600 m from equilibrium, is approximately 0.978 m/s. This was determined using principles of energy conservation. Initial potential energy was converted to kinetic energy to find the speed.
Given a 0.300 kg mass attached to a spring with a force constant (k) of 26.6 N/m, we can determine the speed when the mass is 0.0600 m from equilibrium using energy conservation principles. Initially, the mass is pulled 0.120 m from equilibrium and released.
We start by calculating the total mechanical energy of the system at the maximum displacement (0.120 m):
Potential Energy (PEmax) = 1/2 k xmax²
PEmax = 1/2 x 26.6 N/m x (0.120 m)²
PEmax = 1/2 x 26.6 x 0.0144
PEmax = 0.19152 J
At the position where the mass is 0.0600 m from equilibrium, the potential energy is:
PE = 1/2 k x²
PE = 1/2 x 26.6 N/m x (0.0600 m)²
PE = 1/2 x 26.6 x 0.0036
PE = 0.04788 J
The difference in energy will be converted into kinetic energy (KE):
KE = PEmax - PE
KE = 0.19152 J - 0.04788 J
KE = 0.14364 J
Using the kinetic energy formula (KE = 1/2 mv²), we solve for the speed:
0.14364 J = 1/2 x 0.300 kg * v²
0.14364 = 0.150 x v²
v² = 0.14364 / 0.150
v² = 0.9576
v = √0.9576
v ≈ 0.978 m/s
Thus, the speed of the mass when it is 0.0600 m from equilibrium is approximately 0.978 m/s.
Matter is anything that has
Answer:
Space
Explanation:
Hope this helped
Answer:
VOLUME AND MASS IF U ;LOOK AT THE OTHERGUIYS ANSWER IS SPACE WHICH IS CORRECT BUT MY ANSSIGNMENT DOES NOT HAVE THAT ANSWER SO ITS VOLUME AND MASS
Explanation:
Two spheres exert a gravitational attraction on each other. Which of the following changes to this system would reduce the gravitational attraction by one half?
- Decreasing one of the masses to 1/2 of its original value
- Increasing the distance by a factor of [tex]\sqrt{2}[/tex]
Explanation:
There are no options provided, however we can still answer the question.
In fact, the magnitude of the gravitational force between two objects is given by the equation:
[tex]F=G\frac{m_1 m_2}{r^2}[/tex]
where
[tex]G=6.67\cdot 10^{-11} m^3 kg^{-1}s^{-2}[/tex] is the gravitational constant
m1, m2 are the masses of the two objects
r is the separation between them
We notice that:
The gravitational force is proportional to the product between the massesThe gravitational force is inversely proportional to the square of the distance between the massesTherefore, in order to reduce the gravitational attraction by one half, we can do one of the following changes:
- Decreasing one of the masses to 1/2 of its original value: for example, if [tex]m_1'=\frac{1}{2}m_1[/tex], the gravitational force becomes
[tex]F'=G\frac{m_1' m_2}{r^2}=G\frac{\frac{1}{2}m_1m_2}{r^2}=\frac{1}{2}(G\frac{m_1m_2}{r^2})=\frac{1}{2}F[/tex]
- Increasing the distance by a factor of [tex]\sqrt{2}[/tex]: in fact, if [tex]r'=\sqrt{2}r[/tex], the gravitational force becomes
[tex]F'=G\frac{m_1 m_2}{r'^2}=G\frac{m_1m_2}{(\sqrt{2}r)^2}=\frac{1}{2}(G\frac{m_1m_2}{r^2})=\frac{1}{2}F[/tex]
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2 Points
What type of electromagnetic radiation is being used in the picture?
Answer:
Ultraviolet radiation.
A ball is falling after rolling off a tall roof. The ball has what type of energy.
Answer:
Correct answer: Kinetic energy of rotation, kinetic energy of translation and potential energy
Explanation:
Before it began to rotate it possessed potential energy. After being released from the rest it starts to roll on the roof and get kinetic energy of rotation, kinetic energy of translation along with potential energy.
God is with you!!!
A vitamin called folic acid is very important for neural tube development, which is the development of the baby's
Answer:
Explanation:
Folic acid is the form of folate found in vitamin supplements and fortified foods. Fortified foods, also called enriched foods, are foods that have specific nutrients added to them.
Neural tube defect happen when the tissues and bone around the brain and spine do not grow well. Neural tube can happen in the third and fourth week after conception (the first or second week after your first missed period). This could be before you know that you are pregnant.
Folic acid lowers the risk of your unborn baby having a neural tube defect . Neural tube are a group of serious birth defects that affect a baby’s spinal cord, brain and skull. Some babies with severe Neural tube defect birth are stillborn or do not survive long after birth. Spina Bifida is the most common Neural tube defect.
Kinetic energy is the energy of motion. How could you determine how much kinetic energy an object has?
A.
by finding its speed and direction
B.
by finding its mass and speed
C.
by finding its height and width
D.
by finding its position and form
Answer:
B
Explanation:
this is because the formula for calculating 1 half of the mass multiplied by the velocity squared
Answer:
b
Explanation:
Paul the penguin loves to skate on ice. If he glides 40 Meters in 12 seconds toward the north’s what is Paul velocity
Paul the penguin's velocity is 3.33[tex]ms^{-1}[/tex].
Explanation:
Paul glided (displaced) 40 meters towards north.
The time he taken to displace is 12 seconds.
∴displacement d= 40m.
time t= 12s.
The speed of a body in a certain direction is the measure of Velocity.
The velocity of the an moving object is given by ratio of the rate of change in the distance by the time taken.
⇒Velocity v= [tex]\frac{displacement}{time}[/tex].
=[tex]\frac{40}{12}[/tex].
=3.33 [tex]ms{-1}[/tex].
Thus the velocity of the Paul the penguin glided on ice is 3.33 [tex]ms{-1}[/tex].
In this section of a circuit, resistors R1, R2, and R3 combine for an equivalent resistance of 4.0 ohm. What is R2? Let R1 = 6.0 ohm and R3= 2.0 ohm.
Answer:
R2=10ohm
Explanation:
In this combination
R2 and R3 are in series combination where R1 is in parallel combination with them hence
R2+R3=R`
And
1/R`+1/R1=1/Req
1/R`=1/Req-1/R1
1/R`=1/4-1/6
=1/12
R`=12ohm
Also
R2+R3=R`
R2=R`-R3
R2=12-2
R2=10ohm
I need help with this problem.
Answer:
-6, [tex]-5\frac{2}{5}[/tex], [tex]-4\frac{1}{5}[/tex]
Explanation:
The solution of answer is in the picture attached
The olive oil in a bottle has a mass of 360 g. If the density of olive oil is 0.9 g/mL, what is its volume?
A. 300 mL
B. 400 mL
C. 500 mL
D. 600 mL
Answer:
B-400 mL
Explanation:
Volume = Mass divided by density
360 / 0.9 = 400
5. How are humans affecting global climate change?
Answer:
Due to the generation of carbon dioxide
Explanation:
Climate change occurs due to global warming generated by greenhouse gases. Carbon dioxide is the gas that does not leave or allow solar radiation from the sun to rise from the Earth's surface, increasing the average temperature of the planet.
Burning fossil fuels for energy generation such as coal is another source of carbon dioxide generation. The generation of energy should be directed towards the production of renewable energies, using sources such as solar and wind energy.
A softball is thrown straight up. To what height will it go if it takes 2.0 sec to reach the highest point?
Answer:
4.4 m
Explanation:
Given:
v = 0 m/s
a = -9.8 m/s²
t = 2.0 s
Find: Δy
Δy = vt − ½ at²
Δy = 0 − ½ (-9.8) (2.0)²
Δy = 4.4 m
Looking at the image below (a roller coaster model), explain why the potential energy went from 100 (at top) to zero, and the kinetic energy from zero to 100 (joules)?
Answer:
Explanation:
The total energy at all times is equal to the sum of kinetic and potential energy.
Eu = Ep + Ek
If an object has reached a finite height point and is stationary, then the potential energy is maximum (100 J) and the kinetic energy is equals to zero.
If the same object started to fall before the impact on the ground itself, the kinetic energy is maximum (100 J) and the potential energy is minimum ie zero.
God is with you!!!
How can stars form from dust and gasses
Answer:
A is right in my opinion
What is an input force?
the applied force on a system
the net force on a system
the force a simple machine applies to an object
the force a person applies to a simple machine
Answer:
The force a person applies to a simple machine
Explanation:
An input force is that variable that enters a simple machine to perform a particular job by multiplying the value of that input force.
As a typical example we can find a lever where a person introduces an input force, this force multiplies according to the mechanical advantage of the lever resulting in the work of a simple machine with more work.
Answer:
The force a simple machine applies to an system
Explanation:
what force is necessary to accelerate a 1,200kg car at a rate of 35m/s^2
Answer:
42000N
Explanation:
F=mass *acceleration
F= 1200*35
F=42000N
Orange juice has a lower or higher viscosity than chocolate syrup
Answer:
Orange juice has a lower viscosity than chocolate syrup.
Explanation:
Viscosity is the measure we use to determine essentially how thick a fluid is. So, a liquid that has a high viscosity, is thicker than a liquid that has a lower viscosity. Viscosity is also benchmarked against that of water since water is considered to he the least viscous fluid. Therefore, the more "watery" the substance, the less viscous it is.
Its easy to imagine how orange juice would have a lower viscosity. When we imagine, we think about how quickly both orange juice and chocolate syrup would flow if they both were to be spilled. Basically, we would be looking at their "flow rate" where orange juice would wash away quickly while chocolate syrup would tend to stick to the surface and be slower. But lets also look why this happens. This happens because of the size of the particles. Orange juice is diluted and has a lot of water molecules which are small and can move very quickly. Chocolate syrup on the other hand is comprised of molecules that are larger and heavier and therefore take a longer time to move. So a liquid with lower viscosity is "less resistant to flow".
An object has a mass of 1kg on Earth. What is its weight on the moon?
Answer:
the Mass of 1 kg object is same in Earth & Moon.
Explanation:
Weight, on the otherhand does change with location depends on the gravity. so the answer is : Weight of one kilo on the surface of moon is 1.622 N. A body is taken from the center of the Earth to the Moon.
The weight on moon will be 166.66 grams.
We have a object on earth.
We have to identify its weight on moon.
What is weight ?Weight is a force acting on the body directed towards the center of earth and is the product of mass and acceleration due to gravity.
W = mg
According to the question -
Mass on earth = 1 Kg = 1000 grams
Now, the weight on moon is 1/6 of that of weight on earth. Therefore -
W(M) = [tex]$\frac{1}{6}[/tex] W(E)
Therefore -
W(M) = [tex]\frac{1}{6}[/tex] x 1000 = 166.66 grams
Hence, the weight on moon will be 166.66 grams.
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Five 60 ohm resistors are connected in parallel. What is their equivalent resistance?
Answer:
i think the answer is 12 ohms
plz mark me as brainliest :)
Answer:
12 ohms
Explanation:
The five 60 Ohms resistance can be represented as follows:
R1 = 60 Ohms
R2 = 60 Ohms
R3= 60 Ohms
R4 = 60 Ohms
R5 = 60 Ohms
Rt =?
Since they are in parallel connections, the equivalent resistance (Rt) can be calculated as follows :
1/Rt = 1/R1 + 1/R2 + 1/R3 + 1/R4 + 1/R5
1/Rt = 1/60 + 1/60 + 1/60 + 1/60 + 1/60
1/Rt = ( 1+1+1+1+1)/60
1/Rt = 5/60
1/Rt = 1/12
Cross multiply to express in linear form
Rt = 12 ohms
Therefore the equivalent resistance is 12ohms
A solution in which [H +] = 10-8 M has a pH of ____ and is
A. 8, basic
B. 6, basic
C. -6,basic
D. -8, acidic
Yo sup??
To solve this question we must learn that
pH=-log([H+])
in the question its given that
[H+]=10^(-8)
therefore
pH=-log(10^(-8))
=-(-8)log10
=8
Since pH>7 therefore its basic
Hence the correct answer is option A ie
A.8, basic
Hope this helps.
Answer: 8 basic
Explanation:
A sphere of mass m" = 2 kg travels with a velocity of magnitude υ") = 8 m/s toward a sphere of mass m- = 3 kg initially at rest, as shown in the figure. The collision of the two spheres is elastic and the length of the rope is l = 4 m. a) Apply the conservation of the momentum for the system of the two spheres and calculate the speed of the sphere m- just after the collision. b) Calculate the maximum height of the mass m-. c) What is the value of the angle θ at the maximum height of the mass m-? d) If the sphere m- is instantaneously at rest at its maximum height, calculate the tension in the string. e) When the height of the sphere of mass m- from its lowest position is 1 m, calculate its speed. f) What is the tension in the string when the sphere of mass m2 is 1 m above its lowest position?
a) 6.4 m/s
b) 2.1 m
c) [tex]61.6^{\circ}[/tex]
d) 14.0 N
e) 4.6 m/s
f) 37.9 N
Explanation:
a)
Since the system is isolated (no external forces on it), the total momentum of the system is conserved, so we can write:
[tex]p_i = p_f\\m_1 u_1 = m_1 v_1 + m_2 v_2[/tex]
where:
[tex]m_1 = 2 kg[/tex] is the mass of the 1st sphere
[tex]m_2 = 3kg[/tex] is the mass of the 2nd sphere
[tex]u_1 = 8 m/s[/tex] is the initial velocity of the 1st sphere
[tex]v_1[/tex] is the final velocity of the 1st sphere
[tex]v_2[/tex] is the final velocity of the 2nd sphere
Since the collision is elastic, the total kinetic energy is also conserved:
[tex]E_i=E_k\\\frac{1}{2}m_1 u_1^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2[/tex]
Combining the two equations together, we can find the final velocity of the 2nd sphere:
[tex]v_2=\frac{2m_1}{m_1+m_2}u_1=\frac{2(2)}{2+3}(8)=6.4 m/s[/tex]
b)
Now we analyze the 2nd sphere from the moment it starts its motion till the moment it reaches the maximum height.
Since its total mechanical energy is conserved, its initial kinetic energy is entirely converted into gravitational potential energy at the highest point.
So we can write:
[tex]KE_i = PE_f[/tex]
[tex]\frac{1}{2}mv^2 = mgh[/tex]
where
m = 3 kg is the mass of the sphere
v = 6.4 m/s is the initial speed of the sphere
[tex]g=9.8 m/s^2[/tex] is the acceleration due to gravity
h is the maximum height reached
Solving for h, we find
[tex]h=\frac{v^2}{2g}=\frac{(6.4)^2}{2(9.8)}=2.1 m[/tex]
c)
Here the 2nd sphere is tied to a rope of length
L = 4 m
We know that the maximum height reached by the sphere in its motion is
h = 2.1 m
Calling [tex]\theta[/tex] the angle that the rope makes with the vertical, we can write
[tex]h = L-Lcos \theta[/tex]
Which can be rewritten as
[tex]h=L(1-cos \theta)[/tex]
Solving for [tex]\theta[/tex], we can find the angle between the rope and the vertical:
[tex]cos \theta = 1-\frac{h}{L}=1-\frac{2.1}{4}=0.475\\\theta=cos^{-1}(0.475)=61.6^{\circ}[/tex]
d)
The motion of the sphere is part of a circular motion. The forces acting along the centripetal direction are:
- The tension in the rope, T, inward
- The component of the weight along the radial direction, [tex]mg cos \theta[/tex], outward
Their resultant must be equal to the centripetal force, so we can write:
[tex]T-mg cos \theta = m\frac{v^2}{r}[/tex]
where r = L (the radius of the circle is the length of the rope).
However, when the sphere is at the highest point, it is at rest, so
v = 0
Therefore we have
[tex]T-mg cos \theta=0[/tex]
So we can find the tension:
[tex]T=mg cos \theta=(3)(9.8)(cos 61.6^{\circ})=14.0 N[/tex]
e)
We can solve this part by applying again the law of conservation of energy.
In fact, when the sphere is at a height of h = 1 m, it has both kinetic and potential energy. So we can write:
[tex]KE_i = KE_f + PE_f\\\frac{1}{2}mv^2 = \frac{1}{2}mv'^2 + mgh'[/tex]
where:
[tex]KE_i[/tex] is the initial kinetic energy
[tex]KE_f[/tex] is the kinetic energy at 1 m
[tex]PE_f[/tex] is the final potential energy
v = 6.4 m/s is the speed at the bottom
v' is the speed at a height of 1 m
h' = 1 m is the height
m = 3 kg is the mass of the sphere
And solving for v', we find:
[tex]v'=\sqrt{v^2-2gh'}=\sqrt{6.4^2-2(9.8)(1)}=4.6 m/s[/tex]
f)
Again, since the sphere is in circular motion, the equation of the forces along the radial direction is
[tex]T-mg cos \theta = m\frac{v^2}{r}[/tex]
where
T is the tension in the string
[tex]mg cos \theta[/tex] is the component of the weight in the radial direction
[tex]m\frac{v^2}{r}[/tex] is the centripetal force
In this situation we have
v = 4.6 m/s is the speed of the sphere
[tex]cos \theta[/tex] can be rewritten as (see part c)
[tex]cos \theta = 1-\frac{h'}{L}[/tex]
where in this case,
h' = 1 m
L = 4 m
And [tex]r=L=4 m[/tex] is the radius of the circle
Substituting and solving for T, we find:
[tex]T=mg cos \theta + m\frac{v^2}{r}=mg(1-\frac{h'}{L})+m\frac{v^2}{L}=\\=(3)(9.8)(1-\frac{1}{4})+(3)\frac{4.6^2}{4}=37.9 N[/tex]