Basic Science - Class 6 General Knowledge - Extra Questions

$$v^{2} - \dfrac{v_{1}^{2}}{4} = 2 g (4.5)$$       (For top) 

$$v_{1}^{2} - v^{2} = 2g (4.5)$$      (For bottom) 

$$v^{2} - \left  (\dfrac{v^{2} + 2g (4.5)}{4}  \right  ) = 2g (4.5)$$  

$$\dfrac{3 v^2}{4} = \left  (\dfrac{5}{4}  \right  ) (2g) (4.5)$$  

$$v^{2} = \dfrac{450}{3} = 150$$  

$$v = \sqrt{150} m/s$$  

Maximum  height from tower
$$v^{2} = (2g) h'$$  

$$h' = \dfrac{v^2}{2g}$$  

$$= \dfrac{v^2}{20}$$  

$$h' = 7.5$$
Height of ball $$= 40 + 7.5 = 47.5$$

Answer 
A molecule is composed of two or more elements that are covalently bonded.
A formula unit indicates the lowest whole number ratio of ions in an ionic compound.
Explanation :
A molecule is composed of two or more elements that are covalently bonded. It is the smallest particle of a covalent substance that has all of the properties of that substance, and it is the smallest particle of that substance that can participate in a chemical reaction. The following diagram shows the formation of four different molecules through covalent bonding.
A formula unit indicates the lowest whole number ratio of ions in an ionic compound. Because the positively and negatively charged ions in an ionic compound are arranged in a crystal lattice, there is no discrete particle comparable to a molecule.

Momentum, product of the mass of a particle and its velocity. Momentum is a vector quantity; i.e., it has both magnitude and direction. Isaac Newton's second law of motion states that the time rate of change of momentum is equal to the force acting on the particle

For every action, there is an equal and opposite reaction
Explanation:
Newton's $$3rd$$ Law states: For every action, there is an equal and opposite reaction.
Remember:
According to this law, forces always act in equal by opposite pairs.
Action and reaction force pairs don't cancel each other out because they act on different objects.
The downward force is the action force. The reaction force is the force that is exerted.
Looking at the picture below, we see that when the force of the finger is against the wall, the force exerted by the wall is pressing back towards the finger.

It is the rotational effect of a force, it is equal to the force multiplied by the perpendicular distance between a pivot and the force.
A moment is the name for the turning effect that forces exert on objects. For example imagine pushing a door open. You push on the door handle and the door rotates around its hinges (the hinges are a pivot). You exerted a force that caused the door to rotate – the rotation was the result of the moment of your pushing force.
Pushing a door open is a very helpful application of moments to think about. Think about the location of the door handle – it is on the opposite side of the door to the hinges. The reason for that is that the moment of a force is related to the size of the force and the size of the perpendicular distance between the force and pivot. The larger the perpendicular distance the larger the turning effect (moment).
If you try to push a door open close to the hinges you will need a considerably larger force!
More about moments
In the diagram below there are two forces: $$F1$$ and $$F2$$ . We can find the moments of both forces by taking moments about the points where the other force acts - i.e. we treat one force as a "pivot" in order to find out the moment of the other.
The moment due to force $$F1$$ .
Take moment about the point where $$F2$$ acts. Moment $$=F1.d$$
The moment due to force $$F2$$
Take moments about the point where $$F1$$ acts. Note $$F2$$ is not perpendicular to the distance, d. In this case we either need to determine the component of the force which is perpendicular to the distance, or we need to determine the component of the distance which is perpendicular to the line of action of the force. In this case we will use the former method, and use the vertical component of $$F2$$ $$(F2_{V})$$ . Moment $$=F2_{V}.d$$

The Atomic mass number.

Explanation:

In an atom, almost all of its mass is contributed by my protons and neutrons. Electrons do not contribute much to the mass of that atom because its mass is negligible compared to that of protons and neutrons.

The mass of protons and neutrons are almost equal are taken to be approximately $$1$$ amu (atomic mass unit).

So the mass of an atom is the mass of the protons and neutrons in it.

This is why the sum of the number of protons and neutrons in an atom equals its atomic mass number.

This means that the velocity over a certain period of time is the instantaneous change in position (dx) over the instantaneous change in time (dt) . This period of time is intentionally very small, hence "instantaneous".

The decomposers are heterotrophic microbes such as bacteria, fungi etc. They decompose the the organic and help recycling of nutrints. thanks

Nucleons are the protons and neutrons in the nucleus  of an atom. Protons are positively charged particles, and neutrons have no charge. Protons are slightly less massive than neutrons. The nucleons make up most of the mass of an atom.

Some activities include urbanization, deforestation and vegetation clearing and reservoirs.
For example, water reservoirs affect the overall amount of water going back into the ocean and deforestation changes water runoff patterns that can affect the water cycle.
Contaminants and pollution from factories and such can also affect water quality.

The distance between the earth and the sun is 149.6 million kilometres (that's 92.95 million miles). Scientists calculate that the average time it takes for light to reach the earth from the sun is 8 minutes and 20 seconds.

A hydrogen atom has

$$1$$ proton
$$1$$ electron
$$0$$ neutrons

For $$B, Z = 5$$ , i.e. its nucleus contains $$5$$ protons, $$5$$ positively charged, massive particles. And if there $$5$$ protons, the neutral atom has $$5$$ electrons conceived to whiz about the nuclear core, with a formal $$1s^{2}\ 2s^{2}\ 2p^{1}$$ configuration.

$$Z$$ refers to the number of (nuclear) protons. The Mg nucleus contains $$12$$ positively charged particles (how can I make this statement?). To balance this charge, (the metal is electrically neutral after all), there must be $$12$$ negatively charged particles associated with the protons; these are called electrons and they are conceived to orbit around the positively charged nucleus.

But the negatively charged particles have negligible mass. How do we account for the mass number of $$25$$ ? Well, there must be $$13$$ particles of zero charge, neutrons, and these reside in the nucleus.

Oxygen has $$8$$ protons, and $$8$$ electrons.

The number of protons of any given atom/element can be determined by the atomic number of the specific element, which is found on the periodic table.

As can be seen on the image the atomic number of Gold, which is the numerical value on top, is $$79$$ . Thus the number of protons in a Gold nucleus is $$79$$ . (Atomic number is always the lowest value of the two numbers, except in the case of Hydrogen where they are the same)

On a side note the number of electrons in a stable uncharged element is also equal to the atomic number, so in this case Gold has $$79$$ electrons.

The number of neutrons in the nucleus of the Gold atom is the mass number of the element (the bottom numerical value) minus the atomic number, so thus the number of neutrons is $$118$$ ( $$196.97$$ was rounded off to $$197$$ ).

So to summarize:
Number of protons is: $$79$$
Number of electrons is: $$79$$ and
Number of neutrons is $$118$$

Neutral atom of sodium contains:

At close distances, a strong nuclear force exists between nucleons. This attractive force comes from the neutrons.

More protons in the nucleus need more neutrons to bind the nucleus together.

The graph below is a plot of the number of neutrons versus the number of protons in various stable isotopes.

The stable nuclei are in the pink band known as the belt of stability.

They have a neutron/proton ratio between $$1:1$$ and $$1.5$$ .

As the nucleus gets bigger, the electrostatic repulsions between the protons gets weaker.

The nuclear strong force is about 100 times as strong as the electrostatic repulsions.

It operates over only short distances.

After a certain size, the strong force is not able to hold the nucleus together.

Adding extra neutrons increases the space between the protons.

This decreases their repulsions but, if there are too many neutrons, the nucleus is again out of balance and decays.

Oxygen has atomic number eight. This means that the nuclei of all oxygen atoms have eight protons. A neutral oxygen atom will also have eight electrons.

In an oxygen ion with a  $$2^{-}$$  charge, the number of protons is still $$8$$ , but the number of electrons is $$10$$ .

This results in the oxygen ion having an octet (8) of valence electrons.

Atomic mass or mass number is defined as number of protons and neutrons present inside the nucleus. In Case of Ag(Silver) it has $$47$$ protons and $$61$$ neutrons. We don't include in definition of mass number . An atom is electrically neutral it means it has same number of protons and electrons so silver has $$47$$ electrons revolving around nucleus

Assuming that we are referring to a stable atom of calcium, we need to find the atomic number of calcium. This can be memorized, or found on the periodic table.

Calcium is the $$20th$$ element, with $$20$$ protons (since the number of protons directly changes the element itself).

Since a stable atom has a net charge of  $$0$$ , we must have $$20$$ electrons.

The number of neutrons will be the same as the number of protons, or else we will have an isotope, in this case, it is also $$20$$ .

For strontium, $$Z = 38$$ . There are $$38$$ positively charged particles in its nucleus. Because the element is neutral, there must be a corresponding number of electrons, negatively charged particles, orbiting the nucleus. It atomic mass is $$87.62\ g \cdot mol^{-1}$$ .
$$^{86}Sr , ^{87}Sr , ^{88} Sr$$ , isotopes ate common. These isotopes MUST have $$48 , 49$$ or $$50$$ neutrons respectively in the nuclei.

Look up each element on the periodic table. The atomic number (usually the whole number....increases one after the other), is the number of protons. Carbon, for example, is atomic number $$6$$ . This means carbon has $$6$$ protons.

Take the mass number for each element above and subtract the atomic number (number of protons). This will give you the number of neutrons.

For example, carbon on periodic table is atomic number $$6$$ . That means $$6$$ protons. Above, it says Carbon- $$12$$ . $$12$$ is the mass number.
Mass number−Atomic number=neutrons
$$12 - 6  = 6$$ neutrons.
Carbon: $$6$$ protons, $$6$$ neutrons....and so on.

One more. Nitrogen- $$20$$
Nitrogen is atomic number $$7$$ . So it has $$7$$ protons.
$$20 - 7 = 13$$ This isotope of nitrogen has $$13$$ neutrons.

It has atomic number $$35$$ , element number $$35$$ on the periodic table, and hence $$35$$ protons in the nucleus of all its atoms.

Kepler's 3rd Law of Planetary Motion says that
$$T^{2} = \dfrac{4 \,.\, \pi^{2} \,.\, r^{3}}{GM}$$
$$T$$ is the period of the orbit. That and the radius of the orbit determine the orbital speed.
The terms $$\dfrac{4 \,.\,\pi^{2}}{G}$$ from that equation are constants. I will assume you want to compare masses without changing radius, so I will assume that the entire expression $$\dfrac{4 \,.\,\pi^{2} \,.\,r^{3}}{G}$$ is to be considered constant.
It is unclear which mass you are asking about. There is the mass of the satellite and the mass of the body being orbited to be considered.
I will first assume you are asking about the mass of the satellite. The mass of the satellite is not part of the expression $$\dfrac{4 \,.\, \pi^{2} \,.\,r^{3}}{GM}$$ . Therefore we can conclude that mass of the satellite does not affect orbital speed.
Now I will assume you are asking about the mass of the body being orbited. The mass of that body is the $$M$$ in the expression $$\dfrac{4 \,.\,\pi^{2} \,.\,r^{3}}{GM}$$ . Therefore if $$M$$ increases, the value of $$T$$ will decrease indicating that the speed has increased.

Oxygen is column $$16$$ of the periodic table along with elements like sulfur. They all have $$6$$ valence electrons in their outer shell making them strong reducers and very good oxidizing agents. In nature, it is almost impossible to find an Oxygen atom with only $$6$$ electrons, as this state is extremely unstable. More likely you will find them bonded ionically or covalently with other atoms or molecules.

Yes iced water is a pure substance.

Explanation:

Both ice and liquid water are the same substance. $$H_2O$$  Though ice water is a mixture of both solid and liquid it is a pure substance, based on the molecular structure of its components .

This is a law because it describes the force but makes not attempt to explain how the force works.

A theory is an explanation of a natural phenomenon. Einstein's General Theory of Relativity explains how gravity works by describing gravity as the effect of curvature of four dimensional spacetime.

Einstein also produced an equation which describes gravity. It may look simple, but in reality it is very complex and very hard to solve.

$$G_{\mu \nu}=\dfrac{8\pi G}{c^4} T_{\mu \nu}$$

The mass number is $$238$$ , so the nucleus has $$238$$ particles in total, including $$92$$ protons.
So, to calculate the number of neutrons we have to subtract:
$$238 - 92 = 146  Answer : The atom has$$ 146$$ neutrons.

The only difference is that the acceleration of the brick on the Moon will be smaller than on the Earth.
Gravity is the effect of the Moon to attract (and be attracted by) objects, with mass, through a FORCE. So, the moon exerts a force on the brick (and viceversa).
This force depends upon the masses of the two objects (and the separation as well) but the Moon has a smaller mass, so the force will be smaller and g on the Moon will be smaller as well!

The instantaneous velocity is the specific rate of change of position (or displacement) with respect to time at a single point (x,t) , while average velocity is the average rate of change of position (or displacement) with respect to time over an interval.

The number of neutrons can be found with the formula:
Number of neutrons = mass number - atomic number
Calcium :
mass number $$= 40.08 \Rightarrow 40$$  
atomic number $$= 20$$  

Number of neutrons = relative atomic mass - atomic number 
Number of neutrons $$= 40 - 20$$
Number of neutrons = $$20$$

Atomic number of phosphorous is $$15$$ . SO , it has $$15$$ electrons and $$15$$ protons.
Mass number of phosphorous is $$31$$

Number of neutrons = Mass number - Number of protons
Number neutrons $$= 31 - 15 = 16$$

It is abiotic because it is not currently alive, or dead, meaning it was, at some point, alive.

New substances are formed when wood burns. Carbon-dioxide and vapors leave the wood, reducing it to ashes and carbon substances.

Let's break the displacement vector in to two perpendicular component i.e the vector which is $$30Km \space  45^o$$ south of west.

So,along west component of this displacement was 30sin45 and along south this was 30cos45

So,net displacement towards west was 80+30sin45=101.20Km

So,net displacement towards west was 80+30sin45=101.20Km

and,towards south it was 30cos45=21.20Km

So,net displacement was $$\sqrt {101.20^2+21.20^2} = 103.4 Km$$

Making an angle of $$tan^- \left ( \dfrac{21.20}{101.20}\right) = 11.82^o$$ w.r.t. west

1,5N downwards

Explanation:

During the entire duration of its flight, the ball is a projectile and the only force acting on it is the force of gravitational attraction of the earth which remains constant at all times at W=mg directed vertically downwards.
(I assume that friction and other effects may be ignored for these purposes

Explanation:
$$v=u+at$$
Use above equation to find time (t)
$$t=\dfrac{v-u}{a}=\dfrac{3-22}{-2.1}=9.05 \space s$$

A) $$1.61$$ seconds

Explanation:
Perhaps the best way to visualise this is on a speed-time graph. To do this, we need to pick apart the question.
When I(Steve J.) edited this answer to correct an error
I did not study or make changes to the graph. Caution.
As a forenote, anything in red relates to the motion of the car and anything in green relates to the mouse.
What do we know?
-the mouse has an initial speed of $$1.3\ kmh^{-1}$$ and has this speed throughout. This is the same as $$\dfrac{13}{36}ms^{-1}$$
- the cat has an itial speed of $$0$$ and accelerates at $$0.45\ ms^{-2}$$
- the cat starts moving immediately, at $$t=0$$ . Yes, the question does say "precisely after", but this is so we don't say that eats the mouse at $$t=0$$ so we're done.
Having said this, we can sketch a graph.
Let $$T$$ be the time taken for the car to catch the mouse
We can use equation of motion to find equation of motion for both the cat and the mouse.
For the cat
$$s=s_{cat}$$
$$u=0$$
$$v= /$$
$$a=0.45$$
$$t=T$$
We can the equation $$s=ut+\dfrac{1}{2}at^{2}$$ to form the equation:
$$s_{cat}=0.0T+\dfrac{1}{2}\times 0.45\ T^{2}$$
$$s_{cat}=0.0T+\dfrac{9}{40}T^{2}$$
For the mouse
$$s=s_{mouse}$$
$$u=\dfrac{13}{36}$$
$$v=\dfrac{13}{36}$$
$$a=0$$
$$t=T$$
Again, $$s=ut+\dfrac{1}{2}at^{2}$$ . Since $$a=0$$ , this simplifies to $$s=ut$$
$$s_{mouse}=\dfrac{13}{36}T$$
Since at $$t=T$$ , the cat and the mouse have the same displacement:
$$s_{mouse}=s_{cat}$$
Solving simultaneously:
$$\dfrac{13}{36}T=0.0T+\dfrac{9}{40}T^{2}$$
$$\dfrac{9}{40}T^{2}-\dfrac{13}{36}T=0$$
$$81T^{2}-130T=0$$
$$T(81 T-130)=0$$
$$T=\dfrac{130}{81}(T\neq 0)$$
So it takes the cat $$1.61$$ seconds to catch up the with the mouse $$(3sf)$$ .

No
Explanation:
For the most part velocity is not zero if an object is accelerating. Since acceleration is the change in velocity over time, there has to be a change in velocity for something to accelerate. In other words, if someone is accelerating, it has to have a variable velocity. If the velocity is constant however, the acceleration is zero ( because the velocity isn't changing over time ).
Although at an instant in time it is possible to have zero velocity whilst accelerating.
For example, if you drop an object at the instant you release it it has zero velocity but it is accelerating. Also if you throw an object upwards it will by subject to an acceleration throughout its flight time, however at the very top of its trajectory it will have zero velocity for an instant in time.
This is the reason that in simple harmonic motion you will notice that when the oscillating object is at maximum displacement it has zero velocity but it is experiencing maximum acceleration. 

Yes.
Explanation:
Momentum is a vector quantity, given by the product of an object's mass and velocity. If the velocity of the object is negative, i.e. the object is travelling in what has been chosen as the negative direction, the momentum will also be negative.
$$\vec p=m. \vec v$$

B) $$0.580$$ metres
Explanation:
Perhaps the best way to visualise this is on a speed-time graph. To do this, we need to pick apart the question.
When I(Steve J.) edited this answer to correct an error
I did not study or make changes to the graph. Caution.
As a forenote, anything in red relates to the motion of the car and anything in green relates to the mouse.
What do we know?
-the mouse has an initial speed of $$1.3\ kmh^{-1}$$ and has this speed throughout. This is the same as $$\dfrac{13}{36}ms^{-1}$$
- the cat has an itial speed of $$0$$ and accelerates at $$0.45\ ms^{-2}$$
- the cat starts moving immediately, at $$t=0$$ . Yes, the question does say "precisely after", but this is so we don't say that eats the mouse at $$t=0$$ so we're done.
Having said this, we can sketch a graph.
Let $$T$$ be the time taken for the car to catch the mouse
We can use equation of motion to find equation of motion for both the cat and the mouse.
For the cat
$$s=s_{cat}$$
$$u=0$$
$$v= /$$
$$a=0.45$$
$$t=T$$
We can the equation $$s=ut+\dfrac{1}{2}at^{2}$$ to form the equation:
$$s_{cat}=0.0T+\dfrac{1}{2}\times 0.45\ T^{2}$$
$$s_{cat}=0.0T+\dfrac{9}{40}T^{2}$$
For the mouse
$$s=s_{mouse}$$
$$u=\dfrac{13}{36}$$
$$v=\dfrac{13}{36}$$
$$a=0$$
$$t=T$$
Again, $$s=ut+\dfrac{1}{2}at^{2}$$ . Since $$a=0$$ , this simplifies to $$s=ut$$
$$s_{mouse}=\dfrac{13}{36}T$$
Since at $$t=T$$ , the cat and the mouse have the same displacement:
$$s_{mouse}=s_{cat}$$
Solving simultaneously:
$$\dfrac{13}{36}T=0.0T+\dfrac{9}{40}T^{2}$$
$$\dfrac{9}{40}T^{2}-\dfrac{13}{36}T=0$$
$$81T^{2}-130T=0$$
$$T(81 T-130)=0$$
$$T=\dfrac{130}{81}(T\neq 0)$$
So it takes the cat $$1.61$$ seconds to catch up the with the mouse $$(3sf)$$ .
For part $$B$$ , we can substitute into either equation, since the distance $$s$$ at the point will be the same for either. I will substitute into the mouse equation.
Let $$T=\dfrac{130}{81}$$
$$s=\dfrac{13}{36}\times \dfrac{130}{81}$$
$$s=\dfrac{1690}{2916}$$
So the mouse (and also the cat) has travelled $$0.580\ m$$ $$(3sf)$$
There number personally seem a but too small, so I'm not sure on my arithmetic, although I think my logic is correct. Hope this helps!.

Part A is easy. The velocity at any time is just the initial velocity plus the product of the acceleration and the time.

$$acceleration=\cfrac{deceleration}{2}$$
so time for acceleration = time for decceleration
$${V}_{max}=4+at=0+5(2t)=10t$$
distance $$=$$ area under v.t graph
$$=\cfrac{1}{2}\times (2t+t)\times {V}_{max}$$
$$1500=\cfrac{3t\times 10t}{2}$$
$${t}^{2}=100$$
$$t=10sec$$
Tota time $$=3t=30sec$$

When wedge is not moving tension in string,
$$a=\frac{(M_{2}\sin\beta-m_{2}\sin\alpha)g}{M_{1}+M_{2}}$$
$$T-M_{1}g\sin\alpha=\dfrac{M_{1}(M_{2}\sin\beta-M_{1}\sin\alpha)}{M_{1}+M_{2}}$$
$$T=\dfrac{M_{1}(M_{2}\sin\beta+M_{1}M_{2}\sin\alpha)}{M_{1}M_{2}}+M_{1}g\sin\alpha$$
$$T=\dfrac{M_{1}M_{2}\sin\beta+M_{1}M_{2}\sin\alpha}{M_{1}+M_{2}}$$
$$T=\dfrac{M_{1}M_{2}}{M_{1}+M_{2}}(\sin\alpha+\sin\beta)$$

In vacuum yes, red and violet light (=single photons or photon packets) travel at the same speed. The frequency of red light is smaller than violet - but the wavelength is longer by precisely the same amount. 

In vertical direction weight are balanced by frictional forces 
As the block are in equilibrium balance forces are in horizontal and vertical direction
For the system of block $$(A + B)$$
$$F = N$$
For block A $$F_{A} = 20\,N$$ and For Block B,
$$f_{B} = f_{A} + 100$$
$$f_{B} = 20 + 100$$
$$f_{B} = 120\,N$$

$$T=mg\cos\theta=\dfrac{mv^2}{r}$$
$$T=0.7 \mathrm{N}$$
$$m=\dfrac{100}{1000}=0.1 \mathrm{kg}$$
$$V=1 \mathrm{m} / \mathrm{s}$$
$$\mu=\dfrac{50}{100}=0.5 \mathrm{m}$$
$$0.7-0.1\times10\times \cos\theta=\dfrac{0.1\times 1\times 1}{0.5}$$
$$0.7-\cos\theta=\dfrac{1}{5}$$
$$\dfrac{7}{10}-\dfrac{1}{5}=\cos \theta$$
$$\dfrac{5}{10}=\cos \theta$$
or $$\cos \theta=\dfrac{1}{2}$$
$$\therefore \theta=60^{\circ}$$

$$\dfrac{mv^2}{R} = \dfrac{25 m}{3} = 8.33 m.$$
$$\mu mg = \dfrac{1}{10}\times10\times m = m$$
$$\mu mg < \dfrac{mv^2}{R}$$
The cyclist will slip while taking the turn.

Answer
$$\dfrac{240 km}{3 hrs} = 80 km / h$$
Ratio $$= \dfrac{64}{80} = \frac{4}{5}$$

MI about COM
$$I = \sum mr^2$$
$$= (0.1\times1^2) + (0.1\times1^2)$$
$$= 0.2 kg m^2$$
MI about one end

$$\overrightarrow{A}+ \overrightarrow{B}= \overrightarrow{B}= \overrightarrow{R}$$

$$A = 20 \ N$$
$$R= 20 \sqrt3$$

$$\overrightarrow{B}= \overrightarrow{R}- \overrightarrow{A}$$

$$B^2= R^2+ A^2 - 2RA\cos 30^\circ$$
$$B^2= 400+ 1200-1200$$
$$B^2= 20 \ N$$

Force of friction arises because of interlocking of irregularities on the two surfaces in contact. When a heavy object is placed on the floor, the interlocking of irregularities on the surfaces of box and floor becomes strong.Thus, to push the heavier box, Seema has to apply a greater force than Iqbal.

Let 
$$OA = \hat{i} + \hat{j} + 2\hat{k}$$
$$OB = \hat{i} + 2 \hat{j} - 2\hat{k}$$
$$\vec{AB} = \vec{OB} - \vec{OA} = \hat{j} - 4\hat{k}$$
$$|AB| = \sqrt{1 + 16} = \sqrt{17}$$
$$\vec{AB} = \dfrac{1}{17} (\hat{j} - 4\hat{k})$$
$$\vec{AP} = (3\hat{i} + 6\hat{j} + 4\hat{k}) - (\hat{i} + \hat{j} - \hat{k})$$
$$= 2\hat{i} + 5\hat{j} + 6\hat{k}$$
$$\vec{w} = 3 \times \dfrac{1}{\sqrt{17}} (\hat{j} - 4\hat{k})$$
$$= \dfrac{3}{\sqrt{17}} (\hat{j} - 4\hat{k}$$
$$v = \vec{w} \times \vec{r}$$
$$= \dfrac{3}{\sqrt{17}} (\hat{j} - 4\hat{k} ) \times (2\hat{i} + 5\hat{j} + 2\hat{k} )$$
$$\Rightarrow \dfrac{3}{\sqrt{17}} \begin{vmatrix}
\hat{i}&  \hat{j}&  \hat{k}\\
0&  1&  -4\\
2&  5&  2\
\end{vmatrix}$$
$$\vec{v} = \dfrac{3}{\sqrt{17}} (22\hat{i} - 8\hat{j} - 2\hat{k})$$

Answer
$$\tan 45^{o} = \dfrac{Vy}{4x} = \dfrac{73.5\sqrt{3}  - 10 t}{73./5} =  1$$
$$73.5 \sqrt{3}  -  10  t  =  73.5$$
$$10 t  = (\sqrt{3} -  1)   ( 73 .5) =  0.73  \times 73 .5$$
$$t  = \dfrac{0.73  \times 73.5}{10} = \dfrac{53 .655}{10}$$
$$t  = 5.36$$

For horizontal equilibrium-

T3​=T1​sinθ

For vertical equilibrium-

T2​=T1​cosθ

Taking ratios-

T2​T3​​=tanθ

And for equilibrium of body T2​=6kgwt=6×10=60N

⟹tanθ=65​

 or θ=tan−165​

Answer
Resistance, read below...
Explanation:
Since Inertia is "The tendency for motion to remain unchanged", this simply means the measurement that an object will keep moving at a constant velocity, direction, speed, or at rest.
As Newton's first Law  states "If the resultant force on a stationary object is 0N, the object will remain stationary. If the resultant force on a moving object is 0N it will carry on moving at the same velocity" (same speed and direction, as velocity, is a vector, which has speed and direction).
Therefore the cause of Inertia is Resistance, as Inertia is the resistance for any moving or stationary object to change its state of motion, or as again previously stated as "The tendency for motion to remain unchanged". Force changes Inertia, as previously stated in Newton's first law if a resultant force is 0N, a stationary object will remain stationary, and a moving object will remain moving with the same velocity, but resistance actually causes it.
Although there is an interesting forum here which goes against my point...

Answer
There is no difference between the two.
Explanation:
Angular frequency , also called radial or circular frequency, measures angular displacement per unit time. Its units are therefore degrees (or radians) per second.
$$\omega  =  2  \pi f$$   or $$\dfrac{2 \pi}{T}$$
where f is frequency of rotation and T time period of rotation.
This is also called Angular velocity as compared to the term Velocity used in linear motion.
Pure translation (fixed direction)         Pure rotation (fixed axis)
Velocity $$v  = \dfrac{dx}{dt}$$   Angular velocity $$\omega  = \dfrac{d \theta}{dt}$$
Other comparable parameters are included in the tables below

Within certain limits, mass does not affect a pendulum's period.

Answer 
(a).
0.064s
(b).
781.2m/s
Explanation:
(a).
To get the time of flight we consider the vertical component of motion:
$$s = \dfrac{1}{2} gt^2$$

$$\therefore t = \sqrt{\dfrac{2s}{g}}$$

$$t = \sqrt{\dfrac{2 \times 0.02}{9.8}}$$

$$t = 0.064 s$$
(b)
The horizontal component of velocity is constant so:
$$v = \dfrac{s}{t} = \dfrac{50}{0.064} = 781 .2m / s$$  

Answer
When the plane is 2mi away from the radar station, its distance's increase rate is approximately 433mi/h.
Explanation:
The following image represents our problem:

P is the plane's positionR is the radar station's positionV is the point located vertically of the radar station at the plane's height
h is the plane's heightd is the distance between the plane and the radar stationx is the distance between the plane and the V point
Since the plane flies horizontally, we can conclude that PVR is a right triangle. Therefore, the pythagorean theorem allows us to know that d is calculated:
$$d = \sqrt{h^2 + x^2}$$
We are interested in the situation when d=2mi, and, since the plane flies horizontally, we know that h=1mi regardless of the situation. We are looking for
$$\dfrac{dd}{dt} = d$$

$$d^2 = h^2 + x^2$$

$$\rightarrow \dfrac{d(d^2)}{dt} = \dfrac{d(d^2)dd}{dd} \dfrac{dd}{dt} = \dfrac{d(h^2)}{dh} - \dfrac{dh}{dt} + \dfrac{d(x^2)}{dx} \dfrac{dx}{dt}$$

$$= 2dd = 2xx$$

$$\rightarrow  d = \dfrac{2xx}{2d} = \dfrac{xx}{d}$$
We can calculate that , when d = 2 mi
$$x = \sqrt{d^2 - h^2} = \sqrt{2^2 - 1^2} = \sqrt{3}$$
mi Knowing that the plane flies at a constant speed of 500mi/h, we can calculate
$$d = \dfrac{\sqrt{3}.500}{2} = 250 \sqrt{3} \approx 433 mi/ h$$  

$$\text{Answer}$$
$$50 m/s$$
$$\text{Explanation:}$$
We're asked to find the speed of an object in free fall one second after it has a speed of $$40 m/s.$$
An object in free fall means that the only force affecting its motion is gravity.
To do this, we can use the equation
$$v_y=v_{0}y+a_{y}t$$
where
$$\bullet v_y$$ is the velocity at time $$t$$ (what we're trying to find)
$$\bullet v_{0}y$$ is the initial velocity ( $$40 m/s$$ in this case, because it's assumed to be travelling downward, which is taken to be the negative y-axis )
$$\bullet a_y$$ is the acceleration, equal to $$-g$$ , which is $$9.81 m/s^2$$
$$\bullet t$$ is the time, which in this case is $$1 s$$
Plugging in known values, we have
$$v_y=40 m/s (9.81m/s^2)(1s)$$
$$=-50 m/s$$
rounded to $$2$$ (or $$1,$$ technically) significant figures.
This the velocity of the object (which is why it includes a sign). The speed is simply the magnitude of this, which is a positive quantity:
$$speed=50m/s$$
You also could have solved this problem neglecting the signs of the initial velocity and acceleration in the above equation:
$$speed=40 m/s +(9.81m/s^2)(1s)=50 m/s$$

Answer 
Five protons and six neutrons.
Explanation:
Isotopes are given by the name of the element and the mass number. Here boron-11 means the name of the element is boron and the mass number is 11.
We are given that boron-10 had five protons in its nucleus, and any element always has the same number of protons in its nucleus (atomic number). So boron-11 has five protons the same as boron-10.
Then the mass number is total protons plus neutrons. For boron-11 this total is 11, and five of the particles are protons, thus 115=6 neutrons.

Answer
When you connect a work of friction to its historical context, you have to take the customs and events into consideration of a time period during which context was written.
The historical events, historical moments which consists of cultural, social, political and economic conditions during certain time period is called Historical context.  
In literature, historical context helps us to understand the writers thought of reaching to a particular point. Different people depend on historical context where it conveys a message with different context which as follows:
For Educators and scholars: It gives information about art, literature,  music, dance and poetry.For Architects and builders: It gives information on design when constructing new structures or buildings and when renovating the existing buildings.  For Judges: It explains the laws, acts and rules.

Velocity