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Question Number 20058 Answers: 1 Comments: 0

What is the difference between ∮ and ∫? Where is ∮ used?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{difference}\:\mathrm{between}\:\oint\:\mathrm{and} \\ $$$$\int?\:\mathrm{Where}\:\mathrm{is}\:\oint\:\mathrm{used}? \\ $$

Question Number 20042 Answers: 0 Comments: 3

In the situation given, all surfaces are frictionless, pulley is ideal and string is light, F = ((mg)/2) , find the acceleration of block 2.

$$\mathrm{In}\:\mathrm{the}\:\mathrm{situation}\:\mathrm{given},\:\mathrm{all}\:\mathrm{surfaces}\:\mathrm{are} \\ $$$$\mathrm{frictionless},\:\mathrm{pulley}\:\mathrm{is}\:\mathrm{ideal}\:\mathrm{and}\:\mathrm{string}\:\mathrm{is} \\ $$$$\mathrm{light},\:{F}\:=\:\frac{{mg}}{\mathrm{2}}\:,\:\mathrm{find}\:\mathrm{the}\:\mathrm{acceleration}\:\mathrm{of} \\ $$$$\mathrm{block}\:\mathrm{2}. \\ $$

Question Number 20040 Answers: 0 Comments: 3

The system shown in figure is given an acceleration ′a′ toward left. Assuming all the surfaces to be frictionless, find the force on the sphere by inclined surface.

$$\mathrm{The}\:\mathrm{system}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{figure}\:\mathrm{is}\:\mathrm{given}\:\mathrm{an} \\ $$$$\mathrm{acceleration}\:'{a}'\:\mathrm{toward}\:\mathrm{left}.\:\mathrm{Assuming} \\ $$$$\mathrm{all}\:\mathrm{the}\:\mathrm{surfaces}\:\mathrm{to}\:\mathrm{be}\:\mathrm{frictionless},\:\mathrm{find} \\ $$$$\mathrm{the}\:\mathrm{force}\:\mathrm{on}\:\mathrm{the}\:\mathrm{sphere}\:\mathrm{by}\:\mathrm{inclined} \\ $$$$\mathrm{surface}. \\ $$

Question Number 20038 Answers: 1 Comments: 1

In the figure shown, m slides on inclined surface of wedge M. If velocity of wedge at any instant be v, find velocity of m with respect to ground.

$$\mathrm{In}\:\mathrm{the}\:\mathrm{figure}\:\mathrm{shown},\:{m}\:\mathrm{slides}\:\mathrm{on} \\ $$$$\mathrm{inclined}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{wedge}\:{M}.\:\mathrm{If}\:\mathrm{velocity} \\ $$$$\mathrm{of}\:\mathrm{wedge}\:\mathrm{at}\:\mathrm{any}\:\mathrm{instant}\:\mathrm{be}\:{v},\:\mathrm{find} \\ $$$$\mathrm{velocity}\:\mathrm{of}\:{m}\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{ground}. \\ $$

Question Number 20035 Answers: 1 Comments: 1

In the following cases, find out the acceleration of the wedge and the block, if an external force F is applied as shown. (Both pulleys and strings are ideal)

$$\mathrm{In}\:\mathrm{the}\:\mathrm{following}\:\mathrm{cases},\:\mathrm{find}\:\mathrm{out}\:\mathrm{the} \\ $$$$\mathrm{acceleration}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wedge}\:\mathrm{and}\:\mathrm{the}\:\mathrm{block}, \\ $$$$\mathrm{if}\:\mathrm{an}\:\mathrm{external}\:\mathrm{force}\:{F}\:\mathrm{is}\:\mathrm{applied}\:\mathrm{as} \\ $$$$\mathrm{shown}.\:\left(\mathrm{Both}\:\mathrm{pulleys}\:\mathrm{and}\:\mathrm{strings}\:\mathrm{are}\right. \\ $$$$\left.\mathrm{ideal}\right) \\ $$

Question Number 20014 Answers: 0 Comments: 1

A person in lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance d of 1.2 m from the person. In the following, state of the lift′s motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II. List I P. Lift is accelerating vertically up Q. Lift is accelerating vertically down with an acceleration less than the gravitational acceleration R. Lift is moving vertically up with constant speed S. Lift is falling freely List II 1. d = 1.2 m 2. d > 1.2 m 3. d < 1.2 m 4. No water leaks out of the jar

Question Number 19986 Answers: 1 Comments: 0

An aeroplane has to go from a point A to point B, 500 km away due 30° east of north. A wind is blowing due north at a speed of 20 ms^(−1) . The air speed of the plane is 150 ms^(−1) . Find the direction in which the pilot should head the plane to reach point B.

$$\mathrm{An}\:\mathrm{aeroplane}\:\mathrm{has}\:\mathrm{to}\:\mathrm{go}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:{A} \\ $$$$\mathrm{to}\:\mathrm{point}\:{B},\:\mathrm{500}\:\mathrm{km}\:\mathrm{away}\:\mathrm{due}\:\mathrm{30}°\:\mathrm{east} \\ $$$$\mathrm{of}\:\mathrm{north}.\:\mathrm{A}\:\mathrm{wind}\:\mathrm{is}\:\mathrm{blowing}\:\mathrm{due}\:\mathrm{north} \\ $$$$\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{20}\:\mathrm{ms}^{−\mathrm{1}} .\:\mathrm{The}\:\mathrm{air}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{plane}\:\mathrm{is}\:\mathrm{150}\:\mathrm{ms}^{−\mathrm{1}} .\:\mathrm{Find}\:\mathrm{the}\:\mathrm{direction} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{the}\:\mathrm{pilot}\:\mathrm{should}\:\mathrm{head}\:\mathrm{the} \\ $$$$\mathrm{plane}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{point}\:{B}. \\ $$

Question Number 19970 Answers: 0 Comments: 4

The velocity-time graph of a body is shown in figure. The displacement covered by the body in 8 seconds is

$$\mathrm{The}\:\mathrm{velocity}-\mathrm{time}\:\mathrm{graph}\:\mathrm{of}\:\mathrm{a}\:\mathrm{body}\:\mathrm{is} \\ $$$$\mathrm{shown}\:\mathrm{in}\:\mathrm{figure}.\:\mathrm{The}\:\mathrm{displacement} \\ $$$$\mathrm{covered}\:\mathrm{by}\:\mathrm{the}\:\mathrm{body}\:\mathrm{in}\:\mathrm{8}\:\mathrm{seconds}\:\mathrm{is} \\ $$

Question Number 19890 Answers: 1 Comments: 0

A man on top of a tower of height 35m throws a stone vertically upwards with a speed of 14m/s. Find: (i)the height above the ground, reached by the stone. (ii)the speed of the stone,when it reaches the ground.

$${A}\:{man}\:{on}\:{top}\:{of}\:{a}\:{tower}\:{of}\:{height} \\ $$$$\mathrm{35}{m}\:{throws}\:{a}\:{stone}\:{vertically} \\ $$$${upwards}\:{with}\:{a}\:{speed}\:{of}\:\mathrm{14}{m}/{s}. \\ $$$${Find}: \\ $$$$\left({i}\right){the}\:{height}\:{above}\:{the}\:{ground}, \\ $$$${reached}\:{by}\:{the}\:{stone}. \\ $$$$\left({ii}\right){the}\:{speed}\:{of}\:{the}\:{stone},{when} \\ $$$${it}\:{reaches}\:{the}\:{ground}. \\ $$

Question Number 19795 Answers: 1 Comments: 0

One morning, each member of Manjul′s family drank an 8-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank 1/7-th of the total amount of milk and 2/17-th of the total amount of coffee. How many people are there in Manjul′s family?

$$\mathrm{One}\:\mathrm{morning},\:\mathrm{each}\:\mathrm{member}\:\mathrm{of}\:\mathrm{Manjul}'\mathrm{s} \\ $$$$\mathrm{family}\:\mathrm{drank}\:\mathrm{an}\:\mathrm{8}-\mathrm{ounce}\:\mathrm{mixture}\:\mathrm{of} \\ $$$$\mathrm{coffee}\:\mathrm{and}\:\mathrm{milk}.\:\mathrm{The}\:\mathrm{amounts}\:\mathrm{of}\:\mathrm{coffee} \\ $$$$\mathrm{and}\:\mathrm{milk}\:\mathrm{varied}\:\mathrm{from}\:\mathrm{cup}\:\mathrm{to}\:\mathrm{cup},\:\mathrm{but} \\ $$$$\mathrm{were}\:\mathrm{never}\:\mathrm{zero}.\:\mathrm{Manjul}\:\mathrm{drank}\:\mathrm{1}/\mathrm{7}-\mathrm{th} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{total}\:\mathrm{amount}\:\mathrm{of}\:\mathrm{milk}\:\mathrm{and}\:\mathrm{2}/\mathrm{17}-\mathrm{th} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{total}\:\mathrm{amount}\:\mathrm{of}\:\mathrm{coffee}.\:\mathrm{How} \\ $$$$\mathrm{many}\:\mathrm{people}\:\mathrm{are}\:\mathrm{there}\:\mathrm{in}\:\mathrm{Manjul}'\mathrm{s} \\ $$$$\mathrm{family}? \\ $$

Question Number 19828 Answers: 1 Comments: 1

The speed of a train is reduced from 80km/h to 40km/h after the application of the brake. (i)how much further would the train travel before coming to rest (ii)assuming the acceleration is kept constant,how long will it take to bring the train to rest after the application of the brakes?

$${The}\:{speed}\:{of}\:{a}\:{train}\:{is}\:{reduced} \\ $$$${from}\:\mathrm{80}{km}/{h}\:{to}\:\mathrm{40}{km}/{h}\:{after} \\ $$$${the}\:{application}\:{of}\:{the}\:{brake}. \\ $$$$\left({i}\right){how}\:{much}\:{further}\:{would}\:{the}\: \\ $$$${train}\:{travel}\:{before}\:{coming}\:{to}\:{rest} \\ $$$$\left({ii}\right){assuming}\:{the}\:{acceleration}\:{is} \\ $$$${kept}\:{constant},{how}\:{long}\:{will}\:{it} \\ $$$${take}\:{to}\:{bring}\:{the}\:{train}\:{to}\:{rest} \\ $$$${after}\:{the}\:{application}\:{of}\:{the}\:{brakes}? \\ $$

Question Number 19774 Answers: 1 Comments: 0

A balloon is ascending vertically with an acceleration of 0.2 ms^(−2) . Two stones are dropped from it at an interval of 2 s. The distance between them when the second stone dropped is (take g = 9.8 ms^(−2) )

$$\mathrm{A}\:\mathrm{balloon}\:\mathrm{is}\:\mathrm{ascending}\:\mathrm{vertically}\:\mathrm{with} \\ $$$$\mathrm{an}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{0}.\mathrm{2}\:\mathrm{ms}^{−\mathrm{2}} .\:\mathrm{Two}\:\mathrm{stones} \\ $$$$\mathrm{are}\:\mathrm{dropped}\:\mathrm{from}\:\mathrm{it}\:\mathrm{at}\:\mathrm{an}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{2}\:\mathrm{s}. \\ $$$$\mathrm{The}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{them}\:\mathrm{when}\:\mathrm{the} \\ $$$$\mathrm{second}\:\mathrm{stone}\:\mathrm{dropped}\:\mathrm{is}\:\left(\mathrm{take}\:{g}\:=\:\mathrm{9}.\mathrm{8}\right. \\ $$$$\left.\mathrm{ms}^{−\mathrm{2}} \right) \\ $$

Question Number 19588 Answers: 0 Comments: 0

Question Number 19511 Answers: 1 Comments: 1

In the arrangement shown, the wedge is smooth and has a mass M. The sphere has a mass m. The system is released from rest from the position shown. There is no friction anywhere. Find the contact force between the wall and the sphere.

$$\mathrm{In}\:\mathrm{the}\:\mathrm{arrangement}\:\mathrm{shown},\:\mathrm{the}\:\mathrm{wedge} \\ $$$$\mathrm{is}\:\mathrm{smooth}\:\mathrm{and}\:\mathrm{has}\:\mathrm{a}\:\mathrm{mass}\:{M}.\:\mathrm{The}\:\mathrm{sphere} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{mass}\:{m}.\:\mathrm{The}\:\mathrm{system}\:\mathrm{is}\:\mathrm{released} \\ $$$$\mathrm{from}\:\mathrm{rest}\:\mathrm{from}\:\mathrm{the}\:\mathrm{position}\:\mathrm{shown}. \\ $$$$\mathrm{There}\:\mathrm{is}\:\mathrm{no}\:\mathrm{friction}\:\mathrm{anywhere}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{contact}\:\mathrm{force}\:\mathrm{between}\:\mathrm{the}\:\mathrm{wall}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{sphere}. \\ $$

Question Number 19509 Answers: 1 Comments: 1

Blocks P and R starts from rest and moves to the right with acceleration a_P = 12t m/s^2 and a_R = 3 m/s^2 . Here t is in seconds. The time when block Q again comes to rest is

$$\mathrm{Blocks}\:{P}\:\mathrm{and}\:{R}\:\mathrm{starts}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{and} \\ $$$$\mathrm{moves}\:\mathrm{to}\:\mathrm{the}\:\mathrm{right}\:\mathrm{with}\:\mathrm{acceleration} \\ $$$${a}_{{P}} \:=\:\mathrm{12}{t}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \:\mathrm{and}\:{a}_{{R}} \:=\:\mathrm{3}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} .\:\mathrm{Here}\:{t} \\ $$$$\mathrm{is}\:\mathrm{in}\:\mathrm{seconds}.\:\mathrm{The}\:\mathrm{time}\:\mathrm{when}\:\mathrm{block}\:{Q} \\ $$$$\mathrm{again}\:\mathrm{comes}\:\mathrm{to}\:\mathrm{rest}\:\mathrm{is} \\ $$

Question Number 19472 Answers: 0 Comments: 3

A fishing boat is anchored 9 km away from the nearest point on the shore. A messanger must be sent from the fishing boat to a camp, 15 km from the point on shore closest to boat. If the messanger can walk at a speed of 5 km per hour and can row at 4 km/h, determine the distance of that point (in km) from the shore, where he must land so as to reach the shore in least possible time.

$$\mathrm{A}\:\mathrm{fishing}\:\mathrm{boat}\:\mathrm{is}\:\mathrm{anchored}\:\mathrm{9}\:\mathrm{km}\:\mathrm{away} \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{nearest}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{shore}.\:\mathrm{A} \\ $$$$\mathrm{messanger}\:\mathrm{must}\:\mathrm{be}\:\mathrm{sent}\:\mathrm{from}\:\mathrm{the}\:\mathrm{fishing} \\ $$$$\mathrm{boat}\:\mathrm{to}\:\mathrm{a}\:\mathrm{camp},\:\mathrm{15}\:\mathrm{km}\:\mathrm{from}\:\mathrm{the}\:\mathrm{point} \\ $$$$\mathrm{on}\:\mathrm{shore}\:\mathrm{closest}\:\mathrm{to}\:\mathrm{boat}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{messanger} \\ $$$$\mathrm{can}\:\mathrm{walk}\:\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{5}\:\mathrm{km}\:\mathrm{per}\:\mathrm{hour} \\ $$$$\mathrm{and}\:\mathrm{can}\:\mathrm{row}\:\mathrm{at}\:\mathrm{4}\:\mathrm{km}/\mathrm{h},\:\mathrm{determine}\:\mathrm{the} \\ $$$$\mathrm{distance}\:\mathrm{of}\:\mathrm{that}\:\mathrm{point}\:\left(\mathrm{in}\:\mathrm{km}\right)\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{shore},\:\mathrm{where}\:\mathrm{he}\:\mathrm{must}\:\mathrm{land}\:\mathrm{so}\:\mathrm{as}\:\mathrm{to} \\ $$$$\mathrm{reach}\:\mathrm{the}\:\mathrm{shore}\:\mathrm{in}\:\mathrm{least}\:\mathrm{possible}\:\mathrm{time}. \\ $$

Question Number 19419 Answers: 1 Comments: 1

sin z=200 find z

$$\mathrm{sin}\:\boldsymbol{{z}}=\mathrm{200} \\ $$$$ \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{z}} \\ $$

Question Number 19362 Answers: 0 Comments: 5

The block Q moves to the right with a constant velocity v_0 as shown in figure. The relative velocity of body P with respect to Q is (assume all pulleys and strings are ideal)

$$\mathrm{The}\:\mathrm{block}\:{Q}\:\mathrm{moves}\:\mathrm{to}\:\mathrm{the}\:\mathrm{right}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{constant}\:\mathrm{velocity}\:{v}_{\mathrm{0}} \:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{figure}. \\ $$$$\mathrm{The}\:\mathrm{relative}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{body}\:{P}\:\mathrm{with} \\ $$$$\mathrm{respect}\:\mathrm{to}\:{Q}\:\mathrm{is}\:\left(\mathrm{assume}\:\mathrm{all}\:\mathrm{pulleys}\:\mathrm{and}\right. \\ $$$$\left.\mathrm{strings}\:\mathrm{are}\:\mathrm{ideal}\right) \\ $$

Question Number 19358 Answers: 1 Comments: 1

In the arrangement shown in figure two beads slide along a smooth horizontal rod. The relation between v and v_0 in given position will be

$$\mathrm{In}\:\mathrm{the}\:\mathrm{arrangement}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{figure} \\ $$$$\mathrm{two}\:\mathrm{beads}\:\mathrm{slide}\:\mathrm{along}\:\mathrm{a}\:\mathrm{smooth} \\ $$$$\mathrm{horizontal}\:\mathrm{rod}.\:\mathrm{The}\:\mathrm{relation}\:\mathrm{between} \\ $$$${v}\:\mathrm{and}\:{v}_{\mathrm{0}} \:\mathrm{in}\:\mathrm{given}\:\mathrm{position}\:\mathrm{will}\:\mathrm{be} \\ $$

Question Number 19355 Answers: 0 Comments: 3

Two blocks are placed on a smooth horizontal surface and connected by a string pulley arrangement as shown. If a force F starts acting on block m_1 , then find the relation between acceleration of both masses and their values

$$\mathrm{Two}\:\mathrm{blocks}\:\mathrm{are}\:\mathrm{placed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{smooth} \\ $$$$\mathrm{horizontal}\:\mathrm{surface}\:\mathrm{and}\:\mathrm{connected}\:\mathrm{by}\:\mathrm{a} \\ $$$$\mathrm{string}\:\mathrm{pulley}\:\mathrm{arrangement}\:\mathrm{as}\:\mathrm{shown}. \\ $$$$\mathrm{If}\:\mathrm{a}\:\mathrm{force}\:{F}\:\mathrm{starts}\:\mathrm{acting}\:\mathrm{on}\:\mathrm{block}\:{m}_{\mathrm{1}} , \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{relation}\:\mathrm{between}\:\mathrm{acceleration} \\ $$$$\mathrm{of}\:\mathrm{both}\:\mathrm{masses}\:\mathrm{and}\:\mathrm{their}\:\mathrm{values} \\ $$

Question Number 19345 Answers: 0 Comments: 0

A thin bi − convex lens rest on a plane mirror . it is found that a point objects placed 20cm above the object coincide with it own image. Determine the position and nature of the image when the object is placed (i) 8cm and (ii) 12 from the lens mirror combinatiom

$$\mathrm{A}\:\mathrm{thin}\:\mathrm{bi}\:−\:\mathrm{convex}\:\mathrm{lens}\:\mathrm{rest}\:\mathrm{on}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{mirror}\:.\:\:\mathrm{it}\:\mathrm{is}\:\mathrm{found}\:\mathrm{that}\:\mathrm{a}\:\mathrm{point} \\ $$$$\mathrm{objects}\:\mathrm{placed}\:\mathrm{20cm}\:\mathrm{above}\:\mathrm{the}\:\mathrm{object}\:\mathrm{coincide}\:\mathrm{with}\:\mathrm{it}\:\mathrm{own}\:\mathrm{image}. \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{position}\:\mathrm{and}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{image}\:\mathrm{when}\:\mathrm{the}\:\mathrm{object}\:\mathrm{is}\:\mathrm{placed} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{8cm}\:\:\mathrm{and}\:\:\left(\mathrm{ii}\right)\:\mathrm{12}\:\:\:\:\mathrm{from}\:\mathrm{the}\:\mathrm{lens}\:\mathrm{mirror}\:\mathrm{combinatiom} \\ $$

Question Number 19341 Answers: 0 Comments: 2

Question Number 19476 Answers: 1 Comments: 0

STATEMENT-1 : The graph between kinetic energy and vertical displacement is a straight line for a projectile. STATEMENT-2 : The graph between kinetic energy and horizontal displacement is a straight line for a projectile. STATEMENT-3 : The graph between kinetic energy and time is a parabola for a projectile.

$$\mathrm{STATEMENT}-\mathrm{1}\::\:\mathrm{The}\:\mathrm{graph}\:\mathrm{between} \\ $$$$\mathrm{kinetic}\:\mathrm{energy}\:\mathrm{and}\:\mathrm{vertical}\:\mathrm{displacement} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{for}\:\mathrm{a}\:\mathrm{projectile}. \\ $$$$\mathrm{STATEMENT}-\mathrm{2}\::\:\mathrm{The}\:\mathrm{graph}\:\mathrm{between} \\ $$$$\mathrm{kinetic}\:\mathrm{energy}\:\mathrm{and}\:\mathrm{horizontal} \\ $$$$\mathrm{displacement}\:\mathrm{is}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{for}\:\mathrm{a} \\ $$$$\mathrm{projectile}. \\ $$$$\mathrm{STATEMENT}-\mathrm{3}\::\:\mathrm{The}\:\mathrm{graph}\:\mathrm{between} \\ $$$$\mathrm{kinetic}\:\mathrm{energy}\:\mathrm{and}\:\mathrm{time}\:\mathrm{is}\:\mathrm{a}\:\mathrm{parabola} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{projectile}. \\ $$

Question Number 20046 Answers: 0 Comments: 3

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Question Number 19223 Answers: 1 Comments: 7

A particle P is sliding down a frictionless hemispherical bowl. It passes the point A at t = 0. At this instant of time, the horizontal component of its velocity is v. A bead Q of the same mass as P is ejected from A at t = 0 along the horizontal direction, with the speed v. Friction between the bead and the string may be neglected. Let t_P and t_Q be the respective times taken by P and Q to reach the point B. Then (a) t_P < t_Q (b) t_P = t_Q (c) t_P > t_Q (d) (t_P /t_Q ) = ((length of at arc ACB)/(length of chord AB))

$$\mathrm{A}\:\mathrm{particle}\:{P}\:\mathrm{is}\:\mathrm{sliding}\:\mathrm{down}\:\mathrm{a}\:\mathrm{frictionless} \\ $$$$\mathrm{hemispherical}\:\mathrm{bowl}.\:\mathrm{It}\:\mathrm{passes}\:\mathrm{the}\:\mathrm{point} \\ $$$${A}\:\mathrm{at}\:{t}\:=\:\mathrm{0}.\:\mathrm{At}\:\mathrm{this}\:\mathrm{instant}\:\mathrm{of}\:\mathrm{time},\:\mathrm{the} \\ $$$$\mathrm{horizontal}\:\mathrm{component}\:\mathrm{of}\:\mathrm{its}\:\mathrm{velocity}\:\mathrm{is} \\ $$$${v}.\:\mathrm{A}\:\mathrm{bead}\:{Q}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{mass}\:\mathrm{as}\:{P}\:\mathrm{is} \\ $$$$\mathrm{ejected}\:\mathrm{from}\:{A}\:\mathrm{at}\:{t}\:=\:\mathrm{0}\:\mathrm{along}\:\mathrm{the} \\ $$$$\mathrm{horizontal}\:\mathrm{direction},\:\mathrm{with}\:\mathrm{the}\:\mathrm{speed}\:{v}. \\ $$$$\mathrm{Friction}\:\mathrm{between}\:\mathrm{the}\:\mathrm{bead}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{string}\:\mathrm{may}\:\mathrm{be}\:\mathrm{neglected}.\:\mathrm{Let}\:{t}_{{P}} \:\mathrm{and}\:{t}_{{Q}} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{respective}\:\mathrm{times}\:\mathrm{taken}\:\mathrm{by}\:{P}\:\mathrm{and} \\ $$$${Q}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{the}\:\mathrm{point}\:{B}.\:\mathrm{Then} \\ $$$$\left({a}\right)\:{t}_{{P}} \:<\:{t}_{{Q}} \\ $$$$\left({b}\right)\:{t}_{{P}} \:=\:{t}_{{Q}} \\ $$$$\left({c}\right)\:{t}_{{P}} \:>\:{t}_{{Q}} \\ $$$$\left({d}\right)\:\frac{{t}_{{P}} }{{t}_{{Q}} }\:=\:\frac{\mathrm{length}\:\mathrm{of}\:\mathrm{at}\:\mathrm{arc}\:{ACB}}{\mathrm{length}\:\mathrm{of}\:\mathrm{chord}\:{AB}} \\ $$

Question Number 19140 Answers: 0 Comments: 9

A racing car travels on a track (without banking) ABCDEFA. ABC is a circular arc of radius 2R. CD and FA are straight paths of length R and DEF is a circular arc of radius R = 100 m. The co-efficient of friction on the road is μ = 0.1. The maximum speed of the car is 50 ms^(−1) . Find the minimum time for completing one round.

$$\mathrm{A}\:\mathrm{racing}\:\mathrm{car}\:\mathrm{travels}\:\mathrm{on}\:\mathrm{a}\:\mathrm{track}\:\left(\mathrm{without}\right. \\ $$$$\left.\mathrm{banking}\right)\:{ABCDEFA}.\:{ABC}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circular} \\ $$$$\mathrm{arc}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{2}{R}.\:{CD}\:\mathrm{and}\:{FA}\:\mathrm{are} \\ $$$$\mathrm{straight}\:\mathrm{paths}\:\mathrm{of}\:\mathrm{length}\:{R}\:\mathrm{and}\:{DEF}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{circular}\:\mathrm{arc}\:\mathrm{of}\:\mathrm{radius}\:{R}\:=\:\mathrm{100}\:\mathrm{m}.\:\mathrm{The} \\ $$$$\mathrm{co}-\mathrm{efficient}\:\mathrm{of}\:\mathrm{friction}\:\mathrm{on}\:\mathrm{the}\:\mathrm{road}\:\mathrm{is}\:\mu\:= \\ $$$$\mathrm{0}.\mathrm{1}.\:\mathrm{The}\:\mathrm{maximum}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{car}\:\mathrm{is} \\ $$$$\mathrm{50}\:\mathrm{ms}^{−\mathrm{1}} .\:\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{time}\:\mathrm{for} \\ $$$$\mathrm{completing}\:\mathrm{one}\:\mathrm{round}. \\ $$

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