The viscous force acting on a sphere is directly proportional to the following parameters:

• the radius of the sphere
• coefficient of viscosity
• the velocity of the object

Mathematically, this is represented as

F∝ηarbvc

Now let us evaluate the values of a, b and c.

Substituting the proportionality sign with an equality sign, we get

F=kηarbvc (1)

 

Here, k is the constant of proportionality which is a numerical value and has no dimensions.

Writing the dimensions of parameters on either side of equation (1), we get

[MLT^–2] = [ML^–1T^–1]^a [L]^b [LT^-1]^c

 

Simplifying the above equation, we get

[MLT^–2] = M^a ⋅ L^–a+b+c ⋅ T^–a–c (2)

 

According to <a href="https://byjus.com/physics/mechanics/">classical mechanics</a>, mass, length and time are independent entities.

Equating the superscripts of mass, length and time respectively from equation (2), we get

a = 1 (3)

–a + b + c = 1 (4)

–a –c = 2 or a + c = 2 (5)

Substituting (3) in (5), we get

1 + c = 2

c = 1 (6)

Substituting the value of (3) & (6) in (4), we get

–1 + b + 1 = 1

b = 1 (7)

Substituting the value of (3), (6) and (7) in (1), we get

F=kηrv

The value of k for a spherical body was experimentally obtained as 6π

Therefore, the viscous force on a spherical body falling through a liquid is given by the equation

F=6πηrv

According to Stokes law, if a spherical body falls into a viscous liquid then the force acting at the interface is proportional to – Radius of the spherical body, velocity of the sphere, and viscosity of this given fluid.

If a body is taken from the surface of the earth to a point at a height ‘h’ above the surface of the earth, then r_i = R and r_f = R + h then,

ΔU = GMm [1/R – 1/(R+h)]

ΔU = GMmh/R(R + h)

When, h<<R, then, R + h = R and g = GM/R².

On substituting this in the above equation we get,

Gravitational Potential Energy ΔU = mgh

When a body of mass (m) is moved from infinity to a point inside the gravitational influence of a source mass (M) without accelerating it, the amount of work done in displacing it into the source field is stored in the form of potential energy this is known as gravitational potential energy. It is represented with the symbol Ug.

The equation for gravitational potential energy is:

⇒ GPE = m⋅g⋅h

Where,

• m is the mass in kilograms,
• g is the acceleration due to gravity (9.8 on Earth)
• h is the height above the ground in meters

G=(m1×m2)/r² This is called universal law of gravitation

Newton’s Law of Universal Gravitation states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them

Any two bodies in the universe attract each other with a force is called Newton universal law of gravitation . It is only attractive force.

Newton’s Law of Universal Gravitation states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

The dimensional formula of Velocity is given by,

M⁰ L¹ T^-1

Where,

• M = Mass
• L = Length
• T = Time

 

Potential energy is defined as the energy stored in an object. Potential energy can be divided into many types; Gravitational potential energy, Elastic Potential energy, Electric Potential Energy etc. Here the gravitational potential energy is defined as the energy possessed by an object by virtue of its position relative to others. Elastic potential energy is defined as the energy possessed by virtue of stresses within its body and an electric potential is defined as the energy possessed by an object by virtue of the total charge stored within.

Potential Energy Formula

The formula for gravitational potential energy is given below.

PE = mgh

Where,

• PE is the potential energy of the object in Joules, J
• m is the mass of the object in kg
• g is the acceleration due to gravity in ms^-2
• h is the height of the object with respect to the reference point in m.

Non uniform motion Eg,while moving in a circular path the speed is constant but direction(velocity)changes continuously

A body can have a constant speed but a changing velocity because the direction can change while the speed is constant. ... However, a body can not have a constant velocity with a changing speed.

For example :

A car can not be slowing down yet still be going the same speed and direction.

Young’s modulus (Y) can be written as
Y = FL / (Ae)
Rearrange the above equation
F = YAe / L
= YAL / L ………[∵ Elongation (e) = L]
= YA
= 2 × 10¹¹ N / m² × [1 cm² × 10^-4 m²/cm²]
= 2 × 10⁷ N

Force required is 2 × 10⁷ N

 

What is torque

What a physics

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The p-block is the region of the periodic table that includes columns 3A to column 8A and does not include helium. There are 35 p-block elements, all of which are in p orbital with valence electrons. The p-block elements are a group of very diverse elements with a wide range of properties.

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The elements s-block and p-block are so-called because their valence electrons are either in an orbital s or p. These are often called Standard Components, in order to differentiate them from the sequence of

</section>

The elements in which the last electron enters the p – orbital of their outermost energy level are called p – block elements. It contains elements of group 13,14, 15, 16, 17 and 18 of the periodic table. General electronic configuration of p – block elements is ns2 np1-6.

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measurement of length in one direction. Examples: width, depth and height are dimensions. ... a square has two dimensions (2D), and a cube has three dimensions (3D). In Physics it can also mean any physical measurement such as length, time, mass, etc.

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Given,

or

escape speed is given as

v = (2GM / R)^1/2

here

M is the mass of the planet

R is the radius of the planet

 

now, for Mars

v_m = (2GM_m / R_m)^1/2

for earth

v_e = (2GM_e / R_e)^1/2 = 11.2 km/s

we are given

M_m = M_e/9

and R_m = R_e/2

thus, we have

v_m = (2GM_m / R_m)^1/2 =  (2G(M_e /9) /  (R_e/2))^1/2

or

v_m = (2/9)^1/2 . (2GM_e / R_e)^1/2 = (2/9)^1/2.v_e

thus,

v_m = 0.4714 x v_e

or

v_m = 0.4714 x 11.2

so, the escape speed of Mars will be

v_m = 5.27 km/s

= 7.47 km/sec.

The impulse experienced by the object equals the change in momentum of the object. In equation form, F • t = m • Δ v. In a collision, objects experience an impulse; the impulse causes and is equal to the change in momentum.

Five limitations of dimensional analysis- 1. The method does not give any information about dimensionless constant K. 2. It fails when a physical quantity depends on more than three physical quantities. 3. It fails When a physical quantity (e.g., s=ut+1/2at square 2) is sum or difference of two or more quantities. 4. It falls to derive relationships which involve trigonometric, logarithm or exponential functions. 5. Sometimes, it difficult to identify the factors on which physical quantities depends.

What is the pascals law?

See in Google

here exists a third category: force of constraint.

Forces of constraint aren't either conservative or non-conservative forces, because they never do work (positive or negative) of their own (only transmit work from other forces). I suppose if you had to classify a force of constraint, it would be conservative, since it doesn't cause any irreversible processes.

Forces of constraint either always act perpendicular to motion, or prevent motion in their direction.

Conservative forces will have an associated potential energy, such that when such a force does positive work on a body, the body changes from having potential energy to kinetic energy. The opposite occurs when a conservative force slows down said object.

Non-conservative forces are forces for which we aren't really interested in tracking associated energy, and even if we were interested it tracking said energy, processes involving non-conservative forces are irreversible, and it is useless to define an associated potential energy.

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Following are the four fundamental forces in nature: Gravitational force: Gravitational force is one of the four fundamental forces of nature. It is the weakest of the four. It is also the attractive force which arises from the gravitational interaction. Example:  (i) The revolving of the moon around the earth is due to gravitational attraction between them. (ii) The formation of tides in the ocean is due to the gravitational force acting between the earth and the moon. Electromagnetic force:: It is the force between charged particles. Charges at rest have electric attraction (between unlike charges) and repulsion (between like charges). Charges in motion produce magnetic force. Together they are called Electromagnetic Force. Strong nuclear force: Strong Nuclear Force: It is the attractive force between protons and neutrons in a nucleus.It is charge-independent and acts equally between a proton and a proton, a neutron and a neutron, and a proton and a neutron. Recent discoveries show that protons and neutrons are built of elementary particles, quarks. Weak nuclear force:  This force appears only in certain nuclear processes such as the β-decay of a nucleus. In β-decay, the nucleus emits an electron and an uncharged particle called neutrino.This particle was first predicted by Wolfgang Pauli in 1931.

Following are the four fundamental forces in nature:

• Gravitational force: Gravitational force is one of the four fundamental forces of nature. It is the weakest of the four. It is also the attractive force which arises from the gravitational interaction.
  Example: 
  (i) The revolving of the moon around the earth is due to gravitational attraction between them.
  (ii) The formation of tides in the ocean is due to the gravitational force acting between the earth and the moon.
• Electromagnetic force:: It is the force between charged particles. Charges at rest have electric attraction (between unlike charges) and repulsion (between like charges). Charges in motion produce magnetic force. Together they are called Electromagnetic Force.
• Strong nuclear force:
• Strong Nuclear Force: It is the attractive force between protons and neutrons in a nucleus.It is charge-independent and acts equally between a proton and a proton, a neutron and a neutron, and a proton and a neutron. Recent discoveries show that protons and neutrons are built of elementary particles, quarks.
• Weak nuclear force:  This force appears only in certain nuclear processes such as the β-decay of a nucleus. In β-decay, the nucleus emits an electron and an uncharged particle called neutrino.This particle was first predicted by Wolfgang Pauli in 1931.

$$\int\limits _{0}^{x} F dx \\ \int\limits _{0}^{x} kxdx \\ k \int\limits _{0}^{x}xdx \\ k \displaystyle [\dfrac{x^2}{2}]^x _0 \\ k \displaystyle [\dfrac{x^2}{2} - \dfrac{0}{2}] \\ \dfrac{kx^2}{2} $$

Work=F.d=(3i+4j-6k).(6i-2j-k) = (3×6)+(4×-2)+(-6×-1) = 18+(-8)+6= 16J Power=work÷time =16÷2 =8 watt

Assume a perfect sphere-shaped planet of radius R and mass M. Now, if a body of mass m is projected from a point A on the surface of the planet. An image is given below for better representation:

sIn the diagram, a line from the center of the planet i.e. O is drawn till A (OA) and extended further away from the surface. In that extended line, two more points are taken as P and Q at a distance of x and dx respectively from the center O.

Now, let the minimum velocity required from the body to escape the planet b

ve

Thus, Kinetic Energy will be

At point P, the body will be at a distance x from the planet’s center and the force of gravity between the object and the planet will be:

To take the body from P to Q i.e. against the gravitational attraction, the work done will be->

Now, the work done against the gravitational attraction to take the body from the planet’s surface to infinity can be easily calculated by integrating the equation for work done within the limits x=R to x=∞.

Thus,

By integrating it further, the following is obtained:

Thus, the work done will be:

Now, to escape from the surface of the planet, the kinetic energy of the body has to be equal to the work done against the gravity going from the surface to infinity. So,

K.E. = W

Putting the value for K.E. and Work, the following equation is obtained:

From this equation, the escape velocity can be easily formulated which is:

Putting the value of g = GM, the value of escape velocity becomes:

From this equation, it can be said that the escape velocity depends on the radius of the planet and the mass of the planet only and not on the mass of the body.

Escape Velocity of Earth:

From the above equation, the escape velocity for any planet can be easily calculated if the mass and radius of that planet are given. For earth, the values of g and R are:

g = 9.8 m

R = 63,781,00 m

So, the escape velocity will be:

ve=2×9.8×63,781,00−−−−−−−−−−−−−−−−√

Escape Velocity of Earth= 11.2 km/s.

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Using pythagoras theorem, resultant displacement is 5 metres due north - east.

5 metres

$$T = 2 \pi m \sqrt{\dfrac{l}{g}}$$

Let us consider, rocket of mass m₀ take off with velocity v₀ from ground. Let the fuel burnt at the rate of dm/dt and burnt gases eject with velocity u w.r.t. rocket.
Let at any instant t, v be the velocity of rocket and m be the mass of rocket. Therefore velocity of burnt gas ejected w.r.t. ground is (v – u). As the fuel burns at the rate of dm/dt, therefore in time dt, dm mass of the fuel will burn and velocity of rocket increases by dv.

Here m is the mass of rocket and dm is the mass of gas ejected. As the fuel burns the mass of rocket decreases. Therefore, to calculate the velocity of rocket at any instant in terms of mass of rocket, dm must be replaced by ‘dm’.