**ROTATIONAL MOTION AND SYSTEM OF PATICLES**

**1.**Three masses 3 kg, 4 kg and 5 kg are located at the corners of an equilateral triangle of side 1m.

Locate the centre of mass of the system.

**. (x, y) = (0.54 m, 0.36 m)****2.**Two particles mass 100 g and 300 g at a given time have velocities 10i – 7j – 3k and 7i – 9j + 6k

ms

^{–1 }respectively. Determine velocity of COM.**Velocity of COM 31-i 34j+ 15k /2 ms**^{–1}**3.**From a uniform disc of radius R, a circular disc of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of original disc. Locate the centre of gravity of the resultant flat body.

**COM of resulting portion lies at**

**R/6 from the centre of the original disc in a direction opposite to the centre of the cut out portion.**

**4.**The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds, (i) What is it angular acceleration (assume the acceleration to be uniform) (ii) How many revolutions does the wheel make during this time?

**α = 4 π rad /s, n = 576**

**5**. A metre stick is balanced on a knife edge at its centre. When two coins, each of mass 5 g are put one on top of the other at the 12.0 cm mark, the stick is found to be balanced at 45.0 cm, what is the mass of the meter stick?

**m = 66.0g**

**6.**A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s. How much work has to be done to stop it?

**4J**

**7.**In the HCI molecule, the separating between the nuclei of the two atoms is about 1.27 A (1A = 10

^{–10}m). Find the approximate location of the CM of the molecule, given that the chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in all its nucleus

**. CM of HCI is**

**located on the line joining H and CI nuclei at a distance of 1.235 Å from the H nucleus.**

**8.**To maintain a rotor at a uniform angular speed of 200 rad s–1, an engine needs to transmit a torque

of
180 Nm. What is the power required by the
engine? Assume that the engine is 100% efficient.

**36 Kw****9.**The motor of an engine is rotating about its axis with an angular velocity of 100 rev/minute. It comes to rest in 15 s, after being switched off. Assuming constant angular deceleration, calculate the number of revolutions made by it before coming to rest.

**12.5 revolutions**.

**10.**

*Starting from rest, a fan takes five seconds to attain the maximum speed of 400 rpm(revolutions*

perminute).Assuming
constant acceleration, find the time taken by the fan in attaining half the

maximum speed

**.***t*= 2.5 s.**11.**A bucket is being lowered down into a well through a rope passing over a fixed pulley of radius 10 cm Assume that the rope does not slip on the pulley. Find the angular velocity and angular acceleration of the pulley at an instant when the bucket is going down at a speed of 20 cm/s and has an acceleration of 4.0 m/s

^{2}.

**2 rad/s,**

**4 rads**

^{- 2}.**12.**A

**uniform sphere of mass 200 g rolls without slipping on a plane surface so that its centre moves at a speed of 2.00 cm/s. Find its kinetic energy.**

**5.6 x 10**

^{-5}J.**13.**The wheel of a motor, accelerated uniformly from rest, rotates through 2.5 radian during the first second. Find the angle rotated during the next second.

**7.5 rad**

**14.**A wheel having moment of inertia 2 kg-m

^{2}about its axis, rotates at 50 rpm about this axis. Find the torque that can stop the wheel in one minute.

**π/18 N.m**

*15***.**A string is wrapped around the rim of a wheel of moment of inertia 0.20 kg-m

^{ 2}and radius 20 cm. The wheel is free to rotate about its axis. Initially, the wheel is at rest. The string is now pulled by a force of 20 N. Find the angular velocity of the wheel after 5.0 seconds.

**100 rad/s.**

**16.**A uniform ladder of mass 10 kg leans against a smooth vertical wall making an angle of 53° with it. The other end rests on a rough horizontal floor. Find the normal force and the frictional force that the floor exerts on the ladder

**98 N, 65 N.**

**17.**A uniform circular disc of mass 200 g and radius 4.0 cm is rotated about one of its diameter at an angular speed of 10 rad/s. Find the kinetic energy of the disc and its angular momentum about the axis of rotation.

**4.0 x 10**

^{-3}J,**8.0x 10**

^{4}J-s.**18.**A wheel rotating at an angular speed of 20 rad/s is brought to rest by a constant torque in 4.0 seconds. If the moment of inertia of the wheel about the axis of rotation is 0.20 kg-m 2, find the work done by the torque in the first two seconds.

**30 J**

**19.**A wheel starts from rest and rotates with constant angular acceleration and reaches an angular speed of 12.0 rad/s in 3.00 s. Find (a) the magnitude of the angular acceleration of the wheel and (b) the angle (in radians) through which it rotates in this time.

**(a) 4.00 rad/s2 (b) 18.0 rad**

**20.**An airliner arrives at the terminal, and its engines are shut off. The rotor of one of the engines has an initial clockwise angular speed of 2 000 rad/s. The engine’s rotation slows with an angular acceleration of magnitude 80.0 rad/s2. (a) Determine the angular speed after10.0 s. (b) How long does it take for the rotor to comes to rest?

**(a) 1200 rad/s (b) 25.0 s**

**21.**A rotating wheel requires 3.00 s to complete 37.0 revolutions. Its angular speed at the end of the 3.00-s interval is 98.0 rad/s. What is the constant angular acceleration of the wheel?

**13.7 rad/s2**

**22.**A racing car travels on a circular track with a radius of 250 m. If the car moves with a constant linear speed of45.0 m/s, find (a) its angular speed and (b) the magnitude and direction of its acceleration

**. (a) 0.180 rad/s (b) 8.10 m/s2 toward the center of the track**

**23.**A disc 8.00 cm in radius rotates at a constant rate of 1 200 rev/min about its central axis. Determine (a) its angular speed, (b) the linear speed at a point 3.00 cm from its center, (c) the radial acceleration of a point on the rim, and (d) the total distance a point on the rim moves in 2.00 s.

**. (a) 126 rad/s (b) 3.78 m/s (c) 1.27 km/s**

^{2}(d) 20.2 m**24.**A bicycle wheel has a diameter of 64.0 cm and a mass of 1.80 kg. Assume that the wheel is a hoop with all of its mass concentrated on the outside radius. The bicycle is placed on a stationary stand on rollers, and a resistive force of 120 N is applied tangent to the rim of the tire. (a) What force must be applied by a chain passing overa 9.00-cm-diameter sprocket if the wheel is to attain an acceleration of 4.50 rad/s

^{2}? (b) What force is required if the chain shifts to a 5.60-cm-diameter sprocket?

**(a) 872 N (b) 1.40 kN**

**25.**A weight of 50.0 N is attached to the free end of a light string wrapped around a reel with a radius of 0.250 m and a mass of 3.00 kg. The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center. The weight is released 6.00 m above the floor. (a) Determine the tension in the string, the acceleration of the mass, and the speed with which the weight hits the floor. (b) Find the speed calculated inpart (a), using the principle of conservation of energy

**(a) 11.4 N, 7.57 m/s**

^{2}, 9.53 m/s down (b) 9.53 m/s**26.**A uniform, thin, solid door has a height of 2.20 m, a width of 0.870 m, and a mass of 23.0 kg. Find its moment of inertia for rotation on its hinges. Are any of the data unnecessary?

**5.80 kg_m**

^{2}; the height makes no difference.**27. A**cylinder of mass 10.0 kg rolls without slipping on a horizontal surface. At the instant its center of mass has a speed of 10.0 m/s, determine (a) the translational kinetic energy of its center of mass, (b) the rotational energy about its center of mass, and (c) its total energy.

**(a) 500 J (b) 250 J (c) 750 J**

**28.**In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius 0.529 x10

^{_10}m around the proton. Assuming that the orbital angular momentum of the electron is equal to

*h*/2, calculate(a) the orbital speed of the electron, (b) the kinetic energy of the electron, and (c) the angular speed of theelectron’s motion

**.**

**(a) 2.19 x10**

^{6}m/s (b) 2.18 x10^{_18}J (c) 4.13 x10^{16}rad/s**GRAVITATION**

**.**

**1**. The mass of planet Jupiter is 1.9 × 10

^{27}kg and that of the sun is 1.99 × 10

^{30}kg. The mean distance of Jupiter from the Sun is 7.8 × 1011m. Calculate gravitational force which sun exerts on Jupiter, and the speed of Jupiter

**.**

**ν = 1.3 × 10**

^{4}m s^{–1}**2.**A mass ‘M’ is broken into two parts of masses m

_{1}, and m

_{2}. How are m

_{1}, and m

_{2}related so that force of gravitational attraction between the two parts is maximum. m

_{1}= m

_{2}= M/ 2

**3.**If the radius of earth shrinks by 2%, mass remaining constant. How would the value of acceleration due to gravity change ?

**increased by 4%**

**4.**Find the value of g at a height of 400 km above the surface of the earth. Given radius of the earth, R = 6400 km and value of g at the surface of the earth = 9.8 ms

^{–2}.

**[Ans. 8.575 ms**]

^{–2}**5.**How far away from the surface of earth does the acceleration due to gravity become 4% of its value on the surface of earth? Radius of earth = 6400 km.

**[Ans. 25,600 km]**

**6.**The gravitational field intensity at a point 10,000 km from the centre of the earth is 4.8 N kg

^{–1}. Calculate gravitational potential at that point

**. –4.8 × 107 J kg–1**

**7.**A geostationary satellite orbits the earth at a height of nearly 36000 km. What is the potential due to earth’s gravity at the site of this satellite (take the potential energy at ∞ to be zero). Mass of earth is 6 × 10

^{24}kg, radius of earth is 6400 km.

**8.**Jupiter has a mass 318 times that of the earth, and its radius is 11.2 times the earth’s radius. Estimate the escape velocity of a body from Jupiter’s surface, given that the escape velocity from the earth’s surface is 11.2 km s

^{–1}.

**36.6 : 1**

**9.**The distance of Neptune and Saturn from the sun is nearly 10

^{13}m and 10

^{12}m respectively. Assuming that they move in circular orbits, then what will be the ratio of their periods.

**10.**Let the speed of the planet at perihelion P in fig be v

_{p}and Sun planet distance SP be r

_{p}Relate (r

_{A}v

_{A}) to thecorresponding quantities at the aphelion (r

_{A}, v

_{A}). Will the planet take equal times to traverse BAC and CPB?

**r**

_{A}>r_{p},**∴**

**ν**

_{p}> ν_{A}**11.**

*Two particles of masses 1.0 kg and 2.0 kg are placed ata separation of 50 cm. Assuming that the only forces acting on the particles are their mutual gravitation, find the initial accelerations of the two particles.*

**2.65 x 10**

^{-10}m/s^{2}**12.**Find the work done in bringing three particles, each having a mass of 100 g, from large distances to the vertices of an equilateral triangle of side 20 cm.

**- 1.0 x 10**.

^{-11}J**13.**Calculate the value of acceleration due to gravity at a point (a) 5.0 km above the earth's surface and (b) 5.0 kmbelow the earth's surface. Radius of earth = 6400 km and the value of g at the surface of the earth is 9.80 m/s

^{2}.

**9.79 m/s**

^{2}.**14.**A satellite is revolving round the earth at a height of 600 km. Find (a) the speed of the satellite and (b) the time period of the satellite. Radius of the earth = 6400 km and mass of the earth = 6 x 10

^{24}kg.

**7.6 km/s**.

**15.**The

*Explorer VIII*satellite, placed into orbit November 3, 1960, to investigate the ionosphere, had the following orbit parameters: perigee, 459 km; apogee, 2 289 km (both distances above the Earth’s surface); and period,112.7 min. Find the ratio

*vp*/

*va*of the speed at perigee to that at apogee.

**1.27**

**16.**A system consists of three particles, each of mass 5.00 g, located at the corners of an equilateral triangle with sides of 30.0 cm. (a) Calculate the potential energy of the system. (b) If the particles are released simultaneously, where will they collide?

**(a) _1.67x 10**

^{_14}J (b) At the center**17.**A satellite of the Earth has a mass of 100 kg and is at an altitude of 2.00 x10

^{6}m. (a) What is the potential energy of the satellite–Earth system? (b) What is the magnitude of the gravitational force exerted by the Earth on the satellite? (c) What force does the satellite exert on the Earth?

**(a) _4.77 x10**

^{9}J (b) 569 N (c) 569 N up**18.**Derive an expression for the work required to move anEarth satellite of mass

*m*from a circular orbit of radius 2

*RE*to one of radius 3

*RE*.

**19.**As an astronaut, you observe a small planet to be spherical. After landing on the planet, you set off, walking always straight ahead, and find yourself returning to your spacecraft from the opposite side after completing a lap of 25.0 km. You hold a hammer and a falcon feather ata height of 1.40 m, release them, and observe that they fall together to the surface in 29.2 s. Determine themass of the planet.

**7.79 x10**

^{14}kg**MECHANICAL PROPERTIES OF SOLIDS AND LIQUIDS**

**1.**An aluminium wire 1m in length and radius 1mm is loaded with a mass of 40 kg hanging vertically. Young’s modulus of Al is 7.0 × 10

^{10}N/m

^{2 }. Calculate (a) tensile stress (b) change in length (c) tensile strain and (d)the force constant of such a wire

**.**

**a)1.27 × 10**

^{8}N/m^{2}b)1.8 × 10^{–3}mc)1.8 × 10^{–3}d)**2.2 10**

^{5}N/m

**2.**The average depth of ocean is 2500 m. Calculate the fractional compression of water at the bottom of ocean, given that the bulk modulus of water is 2.3 × 10

^{9}N/m

^{2}.

**1.08%**

**3.**A force of 5 × 10

^{3}N is applied tangentially to the upper face of a cubical block of steel of side 30 cm. Find the displacement of the upper face relative to the lower one, and the angle of shear. The shear modulus of steel is 8.3 × 10

^{10}pa.

**tan**

^{–1}(0.67 × 10^{–6})**4.**How much should the pressure on one litre of water be changed to compress it by 0.10%.

**2.2 ×10**

^{6}N m^{–2}**5.**Calculate the pressure at a depth of 10 m in an Ocean. The density of sea water is 1030 kg/m3. The atmospheric pressure is 1.01 × 10

^{5}pa.

**2.04 × 10**

^{5}pa**6.**In a hydraulic lift air exerts a force F on a small piston of radius 5cm. The pressure is transmitted to the second

piston of
radius 15 cm. If a car of mass 1350 kg is to be lifted,calculate force F that
is to be applied.

**1470 N**.**7.**How much pressure will a man of weight 80 kg f exert on the ground when (i) he is lying and (2) he is standing on his feet. Given area of the body of the man is 0.6 m

^{2}and that of his feet is 80 cm

^{2}.

**4.9 × 10**

^{4}Nm^{–2}**8.**The manual of a car instructs the owner to inflate the tyres to a pressure of 200 k pa. (a) What is the recommended gauge pressure? (b) What is the recommended absolute pressure (c) If, after the required inflation of the tyres, the car is driven to a mountain peak where the atmospheric pressure is 10% below that at sea level, what will the tyre gauge read?

**a)200K Pa b) 301 k Pa c) 211 k Pa**

**9.**Calculate excess pressure in an air bubble of radius 6mm. Surface tension of liquid is 0.58 N/m.

**387 N m**

^{–2}**10.**Terminal velocity of a copper ball of radius 2 mm through a tank of oil at 20°C is 6.0 cm/s. Compare coefficient of viscosity of oil. Given pcu =8.9 × 10

^{3}kg/m3, ρoil = 1.5 x 10

^{3}kg/m

^{3}

**1.08 kg m**

^{–1}s^{–1}**11.**Calculate the velocity with which a liquid emerges from a small hole in the side of a tank of large cross-sectional area if the hole is 0.2m below the surface liquid (g = 10 ms

^{–2}). 117. A soap bubble of radius 1 cm expands into a bubble of radius 2cm.Calculate the increase in surface energy if the surface tension for soap is 25 dyne/cm.

**2m/s**

**12.**A glass plate of 0.20 m

^{2 }in area is pulled with a velocity of 0.1 m/s over a larger glass plate that is at rest. What force is necessary to pull the upper plate if the space between them is 0.003m and is filled with oil of η = 0.01 Ns/m

^{2}

**1.02 × 10**

^{3}erg

**13.**The area of cross-section of a water pipe entering the basement of a house is 4 × 10

^{–4}m

^{2}. The pressure of water at this point is 3 × 10

^{5}N/m

^{2}, and speed of water is 2 m/s. The pipe tapers to an area of cross section of 2 × 10

^{–4}m

^{2}, when it reaches the second floor 8 m above the basement. Calculate the speed and pressure of water flow at the second floor.

**66.7 × 10**

^{–3}N**14.**A large bottle is fitted with a siphon made of capillary glass tubing. Compare the times taken to empty the bottle when it is filled (i) with water (ii) with petrol of density 0.8 cgs units. The viscosity of water and petrol are 0.01 and 0.02 cgs units respectively.

**2.16 × 10**

^{5 }N/m^{2}**15.**The breaking stress for a metal is 7.8 × 10

^{9}Nm

^{–2}. Calculate the maximum length of the wire made of this metal which may be suspended without breaking. The density of the metal = 7.8 × 10

^{–3}kg m

^{–3}. Take g = 10 Nkg

^{–1}

**10**

^{5}m