Newton's Laws of motion

2) NEWTON ’S LAWS OF MOTION:

1) First Law of Motion:

If the sum of all forces acting on a particle is zero then and only then the particle remains unaccelerated (i.e. remains at rest or moves with constant velocity).

• We can simply put it as if and only if .
• Newton ’s first law is valid only in inertial frames of reference.

Dumb Question:

1) What is an inertial frames of references?

Ans: A non accelerating frame of reference is called an inertial frame of reference. A frame moving with uniform velocity is an inertial frame of reference.

2) Second Law of Motion:

The acceleration of a particle is measured from an inertial frame and is given by the sum of all forces acting on particle divided by its mass.

Here and are measured at same instant.

3) Third Law of Motion:

To every action, There is an equal and opposite reaction. That is if a body A exerts a force on another body B, Then B exerts a force - on A.

Dumb Question:

1) If you apply some force for moving forward on earth, but earth exerts the same opposite force on you, then how do you move?

Ans: An important point about third law is that action and reaction forces act on different bodies. So here the point is that the man exerts some force but not on himself, but on earth and in reaction earth applies same force over him. The mass of earth is very high as compared to man and so the acceleration of earth becomes negligible.

3) FREE BODY DIAGRAM:

A free body diagram consists of a diagrammatic representation of a single body or a sub system of bodies isolated from the surroundings showing all the forces acting on it.

Illustration:

Draw the FBD in following cases:

Fig (1)

Solution:

Dumb Question:

1) Why is the normal acting like this?

Ans: The direction of the normal force is always perpendicular to the surface of contact (common surface).

2) Why is the tension direction like this?

Fig (2)

Ans:

The direction of tension is always away form the point of contact.

Ex: B is pulling A. If we need to draw FBD for both then?

4) ALGORITHM FOR SOLVING PROBLEMS:

Step 1:

Define the system:

On this system you have to apply Newton ’s laws. A system may consist of any no of particles/component but each component must have same acceleration.

Step 2:

Identify the forces:

List out all the forces acting on the system.

Step 3:

Draw FBD of the system.

Step 4:

Choose Ads and write equation. Proper signs must be put with forces or acceleration.

Illustration:

(Example 5.14 page 182 DC Pandey)

In the arrangement shown in figure (3) the strings are light and inextensible. The surface over which blocks are placed is smooth. Find

a) Acceleration of each block

b) The tension in each string

Fig (3)

Solution:
a) Let a be the acceleration of each block and T₁ and T₂ be the tensions in the two strings as shown in figure (4). Taking the two blocks and the two strings as the system.

Fig (4)

Using

Or 21 = (4 + 2 + 1) a

a = 3m/s².

b) Free body diagram (showing forces in x direction) of 4kg block and 1Kg block are shown in figure.

Fig (5)

Using

For 1Kg block, F-T₁ = (1) (a).

Or 21 – T₁= (1) (3) = 3

\ T₁ = 21-3 = 18N

For 4Kg block,

T₂ = (4) (a)

T₂ = (4) (3) = 12N

Dumb Question:

1) When taken as one system, the tensions do not come into picture why?

Ans: When all blocks are taken as one system, the tension do not come into picture because they now became internal forces and internal forces do not affect motion.

Mock Paper for AIEEE'07

Syllabus Physics - Heat, SHM and waves Mathematics - Permutation and Combination, Matrices, Binomial theorem, Exponential log series, sequence and series

Q1. The temperature of an ideal gas is increased from 27 °C to 927 °C. The root mean square speed of its molecules becomes (a) twice (b) half (c) four times (d) one fourth Q2. At a given volume and temperature, the pressure of a gas (a) varies inversely as its mass (b) varies inversely as the square of its mass (c) varies linearly as its mass (d) is independent of its mass Q3. A monatomic ideal gas, initially at temperature T1 is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature T2 by releasing the piston suddenly. If L1 and L2 are the lengths of the gas column before and after expansion respectively, then T1 / T2 is given by (a) (L1 / L2 )2/3 (b) (L1 / L2) (c) (L2 / L1 ) (d) (L2 / L1 )2/3 Q4 A cycle tyre bursts suddenly. This represents an (a) isothermal process (b) isobaric process (c) isochoric process (d) adiabatic process Q5. The slope of isothermal and adiabatic curves are related as (a) isothermal curve slope = adiabatic curve slope (b) isothermal curve slope = g (adiabatic curve slope) (c) adiabatic curve slope = g (isothermal curve slope) (d) adiabatic curve slope = (1/2) (isothermal curve slope) Q6. A box contains n molecules of gas. How will the pressure of the gas be affected if the number of molecules is made 2n (a) pressure will decrease (b) pressure will remain unchanged (c) pressure will be doubled (d) pressure will become three times Q7. The pressures P of an ideal gas and its mean kinetic energy per unit volume are related as (a) P = (b) P = E (c) P = (d) P = Q8. If a gas has f degrees of freedom, the ratio of the specific heats g the gas is (a) (b) (c) 1 + (d) Q9. The temperature of gas is produced by (a) the potential energy of its molecules (b) the kinetic energy of its Molecules (c) the attractive force between its molecules (d) the repulsive force between its molecules Q10 A polyatomic gas with f degrees of freedom has a mean energy per molecules given by (a) (b) (c) (d) None of these Q11 The number of translational degrees of freedom for a diatomic gas is (a) 2 (b) 3 (c) 5 (d) 6 Q12 The internal energy U is a unique function of any state because change in U (a) does not depend upon path (b) depends upon path (c) corresponds to an adiabatic process (d) corresponds to an isothermal process Q13. A cube of side 5 cm made of iron, and having a mass of 1500 gm, is heated from 25° C to 400° C. The specific heat for iron is 0.12 cal/gm °C and the coefficient of volume expansion is 3.5 x 10–5 / °C. The change in internal energy of the cube is (atmospheric pressure = 105 N/m2): (a) 320 kJ (b) 282 kJ (c) 141 kJ (d) 423 kJ Q14. One end of a copper rod of length 1.0 m and area of cross-section 10–3 m2 is immersed in boiling water and other end in ice. If the coefficient of thermal conductivity of copper is 92 cal/ms C° and the latent heat of ice is 8 x 104 cal/kg., then the amount of ice which melt in one minute is: (a) 9.2 x 10–3 kg (b) 8 x 103 kg (c) 6.9 x 10–3 kg (d) 5.4 x 10–3 kg Q15. A tap supplies water at 10 °C and another tap at 100 °C. How much hot water must be taken so that we get 20 kg water at 25°C? (a) 7.2 kg (b) 10 kg (c) 5.6 kg (d) 14.4 kg Q16. The average kinetic energy per mole of hydrogen at a given temperature is: (a) equal to that of helium (b) 3/5 times that of helium (c) 5/3 times that of helium (d) times that of helium Q17. A closed compartment containing gas is moving with some acceleration in horizontal direction. Neglect effect of gravity. Then the pressure in the compartment is: (a) same everywhere (b) lower in the front side (c) lower in the rear side (d) lower in the upper side Q18. The volume of air increases by 5 % in its adiabatic expansion. The percentage decrease in its pressure will be: (a) 5 % (b) 6 % (c) 7 % (d) 8 % Q19. Two spherical vessels of equal volume are connected by a narrow tube. The apparatus contains an ideal gas at one atmosphere and 300 K. Now if one vessel is sphere and 300 K. Now if one vessel is immersed in a bath of constant temperature 600 K and the other in a bath of constant temperature 300 K, then the common pressure will be: (a) 1 atm (b) atm (c) atm (d) atm Q20 A wall has two layers A and B, each made of different material. Both the layers have same thickness. The thermal conductivity of A is twice that of B. Under thermal equilibrium, the temperature difference across the wall is 36 °C. The temperature difference across the layer A is: (a) 24 °C (b) 18 °C (c) 12 °C (d) 6 °C Q21 Which of the following quantities are always zero in a simple harmonic motion ? (a) (b) (c) (d) all of these Q22 Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel. the motion of the particle is (a) Simple harmonic (b) Parabolic (c) Oscillatory (d) non − Periodic Q23 The motion of a particle is given by x = A sin wt + B cos wt. The motion of the particle is (a) Not simple harmonic (b) Simple harmonic with amplitude (c) Simple harmonic with amplitude (A+ B)/2 (d) Simple harmonic with amplitude Q24 When a sound wave goes from one medium to the quantity that remains unchanged is (a) Frequency (b) Amplitude (c) Wavelength (d) Speed Q25 The differential equation of a wave is (a) d2y/dt2 = v2d2y/dx2 (b) d2y/dx2 = v2d2y/dt2 (c) d2y/dx2 = d2y/dt2 (d) d2y/dx2 = − vd2y/dt2 Q26 The relation between velocity of sound in a gas (v) and r.m.s. velocity of molecules of gas (Vrms.) is (a) v = Vrms (g/3)1/2 (b) Vrms = v(2/3)1/2 (c) v = vrms (d) v = Vr.m.s. (3/g)1/2 Q27 When a wave is reflected from a denser medium, the change in phase is (a) 0 (b) p (c) 2 p (d) 3 p Q28 A closed pipe has certain frequency. Now its length is halved. Considering the end correction, its frequency will now become (a) Double (b) More than double (c) Less than double (d) Four times Q29 The fundamental frequency of a closed end organ pipe is n. Its length is doubled and radius is halved. Its frequency will become nearly (a) n/2 (b) n/3 (c) n (d) 2n Q30 The equation of a plane progressive wave is y = 0.9 sin 4 p when it is reflected at a rigid support, its amplitude becomes of its previous value. The equation of the reflected wave is (a) y = 0.6 sin 4p (b) y = − 0.6 sin 4p (c) y = − 0.9 sin 8p (d) y = − 0.6 sin 8p Q31 Three transverse waves are represented by y1 = A cos (kx ‑ wt) y2 = A cos (kx + wt) y3 = A cos (ky ‑ wt) The combination of waves which can produce stationary waves is (a) y1 and y2 (b) y2 and y3 (c) y1 and y3 (d) yl, y2 and y3 Q32 If two waves of same frequency and same amplitude, on superposition, produce a resultant disturbance of the same amplitude, the wave differ in phase by (a) p (b) 2 p/3 (c) Zero (d) p/3 Q33 A body of mass 5 gram is executing S.H.M. A point O. with an amplitude of 10 cm, its maximum is 100 cm/s. Its velocity will be 50 cm s−1 at a distance (in cm) (a) 5 (b) 5 (c) 5 (d) 10 Q34 The potential energy of a particle (Ux) executing SHM is given by (a) Ux = (b) Ux = K1x + K2x2 + K3x3 (c) Ux = Ae −bx (d) Ux = a cosntant Q35 A second's pendulum is placed in a space laboratory orbiting around the earth at a height 3 R from the earth's surface where R is earth's radius. The time period of the pendulum will be (a) Zero (b) 2 (c) 4 sec (d) Infinite Q36 The threshold intensity of sound is 10−12 W m−2. What is the intensity level of sound whose intensity is 10−8 W m−2 ? (a) 40 dB (b) 8 dB (c) 12 dB (d) 20 dB Q37 Two sinusoidal plane waves of same frequency having intensities I0 and 4 I0. are traveling in the same direction, the resultant intensity at a point at which waves meet with a phase difference of zero radian is (a) I0 (b) 5 I0 (c) 9 I0 (d) 3 I0 Q38 A child swinging on a swing in sitting position, stands up, then the time period of the swing will (a) increase (b) decrease (c) remains same (d) increases of the child is long and decreases if the child is short Q39 The displacement y of a wave traveling x ‑direction is given by y =10−4 sin metres where x is expressed in metres and t in seconds. The speed of the wave ‑ motion in ms−1 is (a) 300 (b) 600 (c) 1200 (d) 200 Q40 The displacement of a particle varies according to the relation x = 4 (cos pt + sin pt ). The amplitude of the particle is (a) − 4 (b) 4 (c) 4 (d) 8 [ [ Mathematics ] Q81 The maximum number of points into which 4 circles and 4 straight lines intersect is (A) 26 (B) 50 (C) 56 (D) 72 Q82 The number of products that can be formed with 10 prime numbers taken two or more at a time is (A) 210 (B) 210 − 1 (C) 210 − 11 (D) 210 − 10 Q83 The number of rectangles in the following figure is (a) 5 x 5 (b) 5P2 x 5P2 (c) 5C2 x 5C2 (d) None of these Q84 If n(B) = 2 and the number of mappings from A to B which are onto is 30, then number of elements in A is (A) 4 (B) 5 (C) 6 (D) None of these Q85 The number of ways in which 6 red roses and 3 white roses can form a garland so that all the white roses come together is (A) 2170 (B) 2165 (C) 2160 (D) 2155 Q86 We are required to form different words the help of the word INTEGER. Let m1 be the number of words in which I and N are never together and m2 be the number of words which begin with I and end with R, then is equal to (A) 42 (B) 30 (C) 6 (D) Q87 The number of divisors of 3630, which have a remainder of 1 when divided by 4, is (A) 12 (B) 6 (C) 4 (D) None of these Q88 The sum of divisors of 25 . 37 . 53. 72 is (a) 26 . 38 54 73 (b) 26 . 38 . 54 . 73 − 2 . 3 . 5 . 7 (C) 26 . 38 . 54 . 74 − 1 (D) Q89 If in a chess tournament each contestant plays once against each of the others and in all 45 games are played, then the number of participants is (A) 9 (B) 10 (C) 15 (D) None of these Q90 The number of ways of choosing 10 balls from infinite white, red, blue and green balls is (A) 70 (B) 84 (C) 286 (D) 86 Q91 Number of positive unequal integral solutions of the equation x + y + z = 6 is (A) 4! (B) 3! (C) 5! (D) 2 x 4! Q92 20 people are to travel by a double decker bus which can carry 9 in the lower deck and 11 in the upper deck. In how many ways can the party be seated if 4 keep themselves in the lower deck and 5 keep in the upper deck ? (A) (11!)2 x 9! (B) 11P5 . 6P6 (C) (11!)2 (D) None of these Q93 The number of all possible selections of one 10 or more questions from 10 given questions, each question having an alternative is (A) 310 (B) 210 − 1 (C) 310 − 1 (D) 210 Q94 There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to exceed by 66 the number of games that the men played with the women. The number of participants is (A) 6 (B) 11 (C) 13 (D) None of these Q95 There are 4 parcels and 5 post offices. In how many ways can 4 parcels be got registered? (A) 20 (B) 45 (C) 54 (D) 54 − 45 Q96 Six X have to be placed in the squares of adjoining figure such that each row contains at least one X. In how many different ways can this be done ? (A) 28 (B) 27 (C) 26 (D) 25 Q97 On a railway there are 10 stations. The number of types of tickets required in order that it may be possible to book a passenger from every station to every other is (A) (B) 10!2! (C) (D) Q98 How many words can be formed by taking 4 letters at a time out of letters of the word MATHEMATICS? (A) 2500 (B) 2550 (C) 2454 (D) 3000 Q99 There are 10 cages for keeping 10 animals in circus in which 4 cages are so small that 5 animals out of 10 can not enter into them. In how many ways can 10 animals be kept in 10 cages ? (A) 66400 (B) 86400 (C) 96400 (D) 96000 Q100 A letter lock contains 5 rings each marked with four different letters. The number of all possible unsuccessful attempts to open the lock is (A) 625 (B) 1024 (C) 624 (D) 1023 Q101 If , y , be in AP then x, , z will be in (a) AP (b) GP (c) HP (d) None of these Q102 If a, b, c be in AP ., b, c, d are in GP ., and c, d, e are in HP , then a, c, e will be in (a) AP (b) GP (c) HP (d) None of these Q103 Three non − zero real numbers form an AP and the squares of these numbers taken in the same order form a GP then the number of all possible common ratios of the GP is (a) 1 (b) 2 (c) 3 (d) None of these Q104 If pth , qth, rth and sth terms of an AP are in GP then q − q, q − r, r − s are in (a) AP (b) GP (c) HP (d) None of these Q105 The nth term of the series 4, 14, 30, 52, 80, 114 ………. is (a) n2 + n + 2 (b) 3n2 + n (c) 3n2 − 5n + 2 (d) (n + 1)2 Q106 If in the expansion of (1 + x), the coefficients of (2r + 3)th and (r − 1)th terms are equal, then the value of r is (a) 5 (b) 6 (c) 4 (d) 3 Q107 The coefficient of x17 in the expansion of (x − 1) (x − 2) (x − 3) ….. (x − 18) is (a) 342 (b) (c) − 171 (d) 684 Q108 The middle term in the expansion of is (a) 251 (b) 252 (c) 250 (d) None of these Q109 If in the expansion of occurs in the rth term, then (a) r =10 (b) r = 11 (c) r = 12 (d) r = 13 Q110 The value of upto three decimal places is (a) 9.949 (b) 9.958 (c) 9.948 (d) None of these Q111 is equal to (a) (b) (c) log (2x +1) (d) log Q112 The sum of the series log 4 2 − log 8 2 + log 16 2 − …, is (a) e2 (b) log e 2 + 1 (c) log e 3 − 2 (d) 1 − log e 2 Q113 2 log x − log (x + 1) − log (x − 1) is equal to (a) (b) (c) − (d) None of these Q114 The coefficient of xn , where n is a multiple of , in the expansion of log (1 + x + x2) is (a) − (b) (c) (d) None of these Q115 The sum of the series is equal to (a) log e x (b) 2 log e x (c) − log e (x + 1) (d) None of these Q116. If the system of equations x – ky – z = 0, kx – y – z = 0, x + y – z = 0 has a non-zero solution, then the possible values of k are: (a) – 1, 2 (b) 1, 2 (c) 0, 1 (d) – 1, 1 Q117. If A = and B =, then value of a for which A2 = B is: (a) 1 (b) – 1 (c) 4 (d) no real values Q118. If A = and | A3 | = 125 then the value of a is: (a) ± 1 (b) ±2 (c) ± 3 (d) ± 5 Q119. If A = and I = and A–1 = , then the value of c and d are: (a) (– 6, – 11) (b) (6, 11) (c) (– 6, 11) (d) (6, – 11) Q120. If P = and A = and Q = PAPT and x = PTQ2005P then x is equal to: (a) (b) (c) (d)

Answers