Motion of a car on a banked road

In the vertical direction (Y axis)

Ncosϴ = fsinϴ + mg --------------------(i)

In horizontal direction (X axis)

fcosϴ + Nsinϴ = mv²/r  ----------------(ii)

Since we know that f      μ_sN   

For maximum velocity, f = μ_sN

(i)becomes:

       Ncosϴ = μ_sNsinϴ + mg

Or, Ncosϴ - μ_sNsinϴ = mg

Or, N = mg/(cosϴ- μ_ssinϴ)

Put the above value of N in (ii)

μ_sNcosϴ + Nsinϴ = mv²/r 

μ_smgcosϴ/(cosϴ- μ_ssinϴ) + mgsinϴ/(cosϴ- μ_ssinϴ) = mv²/r 

mg (sinϴ + μ_scosϴ)/ (cosϴ - μ_ssinϴ) = mv²/r 

Divide the Numerator & Denominator by cosϴ, we get

v² = Rg (tanϴ +μ_s) /(1- μ_s tanϴ)

v = √ Rg (tanϴ +μ_s) /(1- μ_s tanϴ)

This is the miximum speed of a car on a banked road.

Special case:

When the velocity of the car = v₀ ,

• No f is needed to provide the centripetal force. (μ_s =0)
• Little wear & tear of tyres take place.

             v_o = √ Rg (tanϴ)

 Problem: A circular racetrack of radius 300 m is banked at an angle of 15°. If the coefficient of friction between the wheels of a race-car and the road is 0.2, what is the

 (a) optimum speed of the racecar to avoid wear and tear on its tyres, and

(b) maximum permissible speed to avoid slipping ?

 Solution.

R = 300m

ϴ = 15^o

μ_s  = 0.2

• v_o = √ Rg tanϴ

      = √300 * 9.8 * tan 15^o

      = 28.1 m/s

• v_max = √ Rg (tanϴ +μ_s) /(1- μ_s tanϴ)

             = √ 300 * 9.8 * (0.2 + tan 15^o)/(1-0.2tan15^o)

             = 38.1 m/s

Newton’s second law of motion states that the force exerted by a body is directly proportional to the rate of change of its momentum. For a body of mass ‘m’, whose velocity changes from u to v in time t, when force ‘F’ is applied.

Relation between linear velocity and angular velocity

Let us consider the randomly shaped body undergoing a rotational motion as shown in the figure below. The linear velocity of the particle is related to the angular velocity. While considering the rotational motion of a rigid body on a fixed axis, the extended body is considered as a system of particles moving in a circle lying on a plane that is perpendicular to the axis, such as the center of rotation lies on the axis.

In this figure, the particle P has been shown to rotate over a fixed axis passing through O. Here, the particle represents a circle on the axis. The radius of the circle is the perpendicular distance between point P and the axis. The angle indicates the angular displacement Δθ of the given particle at time Δt. The average angular velocity in the time Δt is Δθ/Δt. Since Δt tends to zero, the ratio Δθ/Δt reaches a limit which is known as the instantaneous angular velocity dθ/dt. The instantaneous angular velocity is denoted by ω.

From the knowledge of circular motion, we can say that the magnitude of the linear velocity of a particle traveling in a circle relates to the angular velocity of the particle ω by the relation υ/ω= r, where r denotes the radius. At any instant, the relation v/ r = ω applies to every particle that has a rigid body.

If the perpendicular distance of a particle from a fixed axis is r_i, the linear velocity at a given instant v is given by the relation,

V_i = ωr_i

Similarly, we can write the expression for the linear velocity for n different particles comprising the system. From the expression, we can say that for particles lying on the axis, the tangential velocity is zero as the radius is zero. Also, the angular velocity ω is a vector quantity which is constant for all the particles comprising the motion.

Let us consider the randomly shaped body undergoing a rotational motion as shown in the figure below. The linear velocity of the particle is related to the angular velocity. While considering the rotational motion of a rigid body on a fixed axis, the extended body is considered as a system of particles moving in a circle lying on a plane that is perpendicular to the axis, such as the center of rotation lies on the axis.

In this figure, the particle P has been shown to rotate over a fixed axis passing through O. Here, the particle represents a circle on the axis. The radius of the circle is the perpendicular distance between point P and the axis. The angle indicates the angular displacement Δθ of the given particle at time Δt. The average angular velocity in the time Δt is Δθ/Δt. Since Δt tends to zero, the ratio Δθ/Δt reaches a limit which is known as the instantaneous angular velocity dθ/dt. The instantaneous angular velocity is denoted by ω.

From the knowledge of circular motion, we can say that the magnitude of the linear velocity of a particle traveling in a circle relates to the angular velocity of the particle ω by the relation υ/ω= r, where r denotes the radius. At any instant, the relation v/ r = ω applies to every particle that has a rigid body.

If the perpendicular distance of a particle from a fixed axis is r_i, the linear velocity at a given instant v is given by the relation,

V_i = ωr_i

Similarly, we can write the expression for the linear velocity for n different particles comprising the system. From the expression, we can say that for particles lying on the axis, the tangential velocity is zero as the radius is zero. Also, the angular velocity ω is a vector quantity which is constant for all the particles comprising the motion.

Let ρ  be the density of water, and h be the height of water in cylinder, then thrust at the bottom 
F=  pressure × area =(hρg)×π² 

Thrust on the walls, F'= average pressure ×  area 

F'(ρgh/2)(2πrh)=πrh²ρgF′ 
When F=F',

then hρg×πr² = πrh²ρg 
or h=r=10cm

Banking of road means raising the outer edge of the road with respect to inner edge so that the road makes an angle with the horizontal.

When a  vehicle moves along a curve, the force of friction provides the necessary centripetal force. But this force has a limit = μmg.When the speed of the vehicle increases beyond this maximum value, the banking of roads is necessary so that the component of normal reaction of the road towards the centre provides the necessary centripetal force.

From the  figure,   for vertical  equilibrium :

Total upwards force = total downwards force

Ncosθ=mg+fsinθmg=Ncosθ−fsinθ...(1)

As the  forces Nsinθ,fcosθ  provide the necessary centripetal force.

rmv2=Nsinθ+fcosθ...(2)

Thus dividing (2) by (1),

mgrmv2=Ncosθ−fsinθNsinθ+fcosθ 

As f=μsN

Thus,

rgv2=cosθ−μssinθsinθ+μscosθ=1−μstanθtanθ+μs=1−tanλtanθtanθ+tanλ=tan(θ+λ)

v=rgtan(θ+λ)              where tanλ=μ

Venturimeter is based on Bernoulli's theorem. It consists of a two truncated tubes connected by a pipe at narrow ends. The pipe connecting the two tubes is called throat as shown in figure.

The venturimeter tube is positioned horizontally and the liquid is made to enter in it at end A and after passing through throat BC, it leaves tube at end D. Let at A, the area of cross-section of tube be a₁, pressure of liquid be P₁ and velocity be v₁ And at the throat, the area of cross-section of tube be a₂ pressure of liquid be P₂ and velocity be v₂

 uniform motion is defined as the motion of an object in which the object travels in a straight line and its velocity remains constant along that line as it covers equal distances in equal intervals of time, irrespective of the duration of the time.

Non- conservative forces are forces which act on a body and the work done on the body by the force is dependent on the path followed. During the motion of a body due to non-conservative force, mechanical energy is dissipated, so we can call the non-conservative forces as dissipative courses.

A non-conservative force is the one for which work done depends on the path. For eg, frictional force. Friction does more work on the block if one slides it along the indirect path across the tabletop. The longer the path, the more work friction does.

187.2 - 63.54 = 123.66

Rounding off to 4 significant digits,

Answer = 123.7

Density of solid decreases as be increase the temperature. Since mass remains the same but volume increases with increase in temperature that results in decrease in density.

Carnot engine is a theoretical thermodynamic cycle proposed by Leonard Carnot. It gives the estimate of the maximum possible efficiency that a heat engine during the conversion process of heat into work and conversely, working between two reservoirs, can possess.  

It is defined as ratio of net mechanical work done per cycle by the gas to the amount of heat energy absorbed per cycle from the source.

Moment of inertia plays the same role as is played by mass in translatory motion. In translatory motion, mass is a measure of inertia, therefore, moment of inertia is a measure of rotational inertia in rotatory motion.

Moment of inertia is a measure of how difficult it is to rotate a particular body about a given axis. significance: Greater the mass concentrated away from the axis, greater the moment of inertia.