- 5 - TEST 1R. This is the repeat version of TEST 1, which was held during Session.

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1 - 5 - TEST 1R This is the epeat vesion of TEST 1, which was held duing Session. This epeat test should be attempted by those students who missed Test 1, o who wish to impove thei mak in Test 1. IF YOU ARE ATTEMPTING TEST 1 (Repeat): CROSS THIS BOX AND INDICATE YES ON THE FRONT (TITLE) PAGE.

2 - 6 - QUESTION 5 [Maks 17] y A paticle executes unifom cicula motion with adius, as shown in the figue. Fom the figue, deive an expession fo the position vecto in tems of, the angula velocity ω, and the time t by pojecting onto the x and y axes. Take the angle θ at t = 0 to be zeo. θ x Fom, deive the velocity v and the acceleation a of the paticle. Expess the esult fo the acceleation in tems of the position vecto. (c) Calculate the scala poduct v. What does this say about the elative oientation of v and? A stone is tied to a sting and is swung in a hoizontal cicle of adius 1.2m at 1.8m above the gound. The sting beaks and the stone tavels a hoizontal distance of 9.0m befoe stiking the gound. What was the centipetal acceleation of the stone while it was in cicula motion? The shadow of a paticle undegoing unifom cicula motion is pojected onto a sceen that is pependicula to the plane of the cicle. Demonstate that the shadow executes simple hamonic motion. QUESTION 6 [Maks 17] A satellite of mass m is in a cicula obit of adius about the Eath with mass M e and adius R e. Deive an expession fo the kinetic enegy of the satellite in tems of the gavitational constant G, m, M e, and. Eath Hence, deive an expession fo the launch speed of the satellite fom Eath equied to place it in this obit. Show that you esult educes to the escape speed fom the Eath when. Two satellites ae in cicula obits about the Eath at adius and ae on a collision couse, as shown in the figue. (c) Deive an expession fo the total mechanical enegy fo the two-satellite-plus-eath system befoe the collision in tems of G, m, Me, and. The satellites collide completely inelastically (the weckage stays in one piece afte the collision).

3 - 7 - What is the total mechanical enegy of the system afte the collision? Descibe the subsequent motion of the weckage. QUESTION 7 [Maks 16] A block is eleased fom a height h on a fictionless incline. At the base of the incline thee is a ough suface of length l and the coefficient of kinetic fiction between the block and the suface is µ k = 0.3. The suface becomes fictionless again befoe a cicula loop of adius which is followed by a sping attached to a wall. The block emains in contact with the suface at all times. The set-up is shown in the figue; diffeent points of the tack ae labelled A, B, C, D. C h A µ k =0.3 B D What is the speed of the block at point A in tems of the height h? What is the speed of the block at point B in tems of the given paametes? (c) At point C the block is at the maximum height inside the loop. (i) What is the speed of the block at this point? Expess you answe in tems of the given paametes. (ii) Daw a fee-body diagam fo the foces acting on the block at point C. (iii) Find the minimum height h such that the block maintains contact with the loop at point C. Take l = 3cm and = 5cm. The block is bought to est beyond point D afte compessing the sping by 3.0cm. Detemine the value fo the sping constant k, if l = 3cm, = 5cm, h = 18cm, and the mass of the block m = 0.025kg.

4 - 8 - QUESTION 8 [Maks 17] A yo-yo consists of two discs of adius R connected by a shaft of adius R 0 aound which sting is wapped (see figue). The yo-yo has a mass m = 0.020kg, R = 3.0cm, and R 0 = 0.30cm. Assume that the otational inetia of the yo-yo about the cente of mass is the same as that of a disc, I = mr 2 /2. The sting is held while the yo-yo is eleased fom est; as the yo-yo falls, the sting unwinds without slipping. sting R R o Daw a fee-body diagam fo the yo-yo. Which of the foces poduces a toque about the cente of mass? Wite down Newton s second law fo the toque about the cente of mass in tems of the foce(s). (c) Wite down Newton s second law fo the foces acting in the vetical diection. Hence, find expessions in tems of m, g (acceleation due to Eath's gavity), R, and R 0 fo (i) the tension in the sting T (ii) the linea acceleation a of the yo-yo. At the instant when the sting is fully unwound, the yo-yo otates with angula velocity ω 0. What is the minimum value fo ω 0 equied fo the yo-yo to climb a distance l = 0.8m up the sting (no slipping)? QUESTION 9 [Maks 17] A fey on a ive is at est elative to the wate. A passenge spots a cocodile at est in the wate l = 8.0m fom the font of the fey. In esponse, the 50 passenges on boad un to the fa end of the fey, tying to get as fa fom the cocodile as possible. The fey has a mass M =3500kg and length L = 30m; its cente of mass lies at the cente of the fey. The aveage mass of a passenge is 75kg. Assume that oiginally the passenges wee evenly distibuted aound the fey and afte the scae they ae all a distance L/2 fom the cente. See the figue. BEFORE passenges AFTER L l cocodile d cocodile fey fey How fa fom the cocodile is the fey (d in the figue) afte the passenges each the end? [Hint: Conside the cente of mass of the fey and passenges.]

The masses m b and velocities v b of the belongings and the angles at which they e thown ae shown in the figue. Calculate the final velocity of the fey with espect to the wate.

5 - 9 - What is the speed of the fey (elative to the wate) afte the passenges each the end? (c) The engine of the fey will not stat. In despeation, the passenges decide to constuct a couple of catapults and thow thei belongings into the ive so that the fey will ecoil. The masses m b and velocities v b of the belongings and the angles at which they e thown ae shown in the figue. Calculate the final velocity of the fey with espect to the wate. m b = 200 kg v b = 20 m/s m b = 200 kg v b = 20 m/s 30 o 30 o A peson on the bank watches in amusement. What is the final speed of the fey accoding to this obseve if the ive moves at a speed 3km/h in the same diection as the fey? QUESTION 10 [Maks 16] A helicopte popelle consists of thee blades, each of length 5.0m and mass 220kg, with the axis of otation though the cente. Calculate the otational inetia of a od of length L and mass M about an axis at the end of the od. You may use the fomula M 2 I = x dx. L (c) (f) 5.0 m Hence, calculate the otational inetia of the popelle. A block of metal of mass m = 25kg is stuck to one of the blades at a distance l = 2.0m fom the cente. Calculate the otational inetia of the block+popelle system. A moto povides a constant toque τ = 7000Nm along the axis of the popelle. Calculate the time it takes fo the popelle to each an angula speed of 2000pm fom est. [pm evolutions pe minute; emembe, 1 evolution= 2π adians.] What distance has the block tavelled in this time? When the angula speed 2000pm is eached, the moto is tuned off. The metal block shifts 50cm futhe fom the axis. Calculate the new angula speed of the popelle.

06 - ROTATIONAL MOTION Page 1 ( Answers at the end of all questions )

06 - ROTATIONAL MOTION Page 1 ( Answers at the end of all questions ) 06 - ROTATIONAL MOTION Page ) A body A of mass M while falling vetically downwads unde gavity beaks into two pats, a body B of mass ( / ) M and a body C of mass ( / ) M. The cente of mass of bodies B and