TOPIC C: ENERGY EXAMPLES SPRING 2019

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1 TOPI : ENERGY EXMPES SPRING 2019 (Tke g = 9.81 m s 2 ). Q1. Find the kinetic energy of: () bullet of mss 20 g moving t 400 m s 1 ; rcing cr of mss 2.5 tonnes trvelling t 300 km hr 1 ; (c) sphericl rindrop of dimeter 2 mm flling t 20 m s 1. (Tke the density of wter s 1000 kg m 3.) Q2. onsider the collision nd subsequent motion in stright line of two prticles with msses 1 kg nd 10 kg nd velocities before collision of v 1 = 15 m s 1 nd v 2 = 4 m s 1, when the coefficient of restitution e took vlues 0, 0.5 nd 1. The pre-collision nd post-collision velocities (denoted by prime) re given in the following tble. 1 kg v 1 (m s 1 ) 10 kg v 2 (m s 1 ) 1 kg (m s 1 ) 10 kg (m s 1 ) e = e = e = () (c) onfirm the post-collision velocities in the tble. lculte the totl kinetic energy of the two-prticle system before collision nd, for ech vlue of e, the totl kinetic energy of the system fter collision. Wht is the effect of inelstic collisions on (i) momentum; (ii) kinetic energy? (Optionl) Find generl expression for the energy loss in the collision in terms of m 1, m 2, v 1, v 2 nd e. Q kg vn hs 150 hp engine (1 hp = W). Neglecting friction, wht is the mximum stedy speed with which this vn cn scend 10 slope? Q4. Greek legend hs it tht Sisyphus ws condemned for ever to roll boulder up mountin, only to wtch it roll bck down gin under its own weight. ssuming tht the boulder hd mss 2000 kg nd he rolled it up 1 in 3 slope t stedy 0.3 m s 1, find the power required during the scent. Q5. For solid rod under xil forces (tension or compression), Hooke s lw is often expressed s stress = Young s modulus strin Use this to deduce the equivlent stiffness nd elstic modulus for uniform rod of length, cross-sectionl re nd Young s modulus E. Mechnics Exmples for Topic (Energy) - 1 Dvid psley

2 Q6. (Exm 2015) Prticles (mss 0.4 kg) nd (mss 1.2 kg) re ttched to point O by light, inelstic strings of length 3 m. Prticle is held to the side with its string tut nd horizontl (see right). It is then relesed from rest, so tht it swings down nd hits prticle, which initilly hngs t rest. The coefficient of restitution between prticles is 0.7. Find: () (c) the speed of prticle immeditely before the collision; the speeds of both prticles immeditely fter the collision; the loss of kinetic energy in the collision. 0.4 kg 3 m 1.2 kg O 3 m 200 mm 30 o 5 N Q7 (Merim nd Krige) The 0.5 kg collr strts from rest t nd slides with negligible friction on the fixed rod in the verticl plne. Determine the velocity v with which the collr strikes end when cted upon by the 5 N force, which is constnt in direction. Neglect the dimensions of the collr. Q8. spring of nturl length 0.6 m is stretched 100 mm by lod of 50 N. Find: () the stiffness (k) of the spring; the elstic modulus (λ) of the spring; (c) the potentil energy stored in the spring. Q9. spring of nturl length 300 mm is ttched to the floor. When mss of 0.15 kg is plced on the top the spring is compressed by 40 mm. The mss is then pushed down further 100 mm nd relesed. () Find the stiffness of the spring. Find the mximum height (bove the floor) to which the mss rises if: (i) it remins ttched to the spring; (ii) it is not ttched to the spring. Q10. weight 30 N hngs in the loop of light inelstic cble. The cble is then secured to wll by spring of stiffness k = 60 N m 1. If the weight is relesed from rest with the spring initilly stretched by 0.1 m, find: () the mximum speed of the weight; the mximum downwrd displcement of the weight. 30 N k=60 N/m Mechnics Exmples for Topic (Energy) - 2 Dvid psley

3 Q11. (Merim nd Krige) frictionless 6 kg cylindricl collr is relesed from rest in the position shown nd drops onto the spring. lculte: () the mximum deflection of the spring; the velocity v of the collr when the spring hs been compressed 50 mm. 500 mm 6 kg k = 12 kn/m - Q12. (Merim nd Krige) flexible but inextensible chin of length is held on smooth tble with n initil overhng s shown. lculte the velocity v with which the chin will leve the tble if relesed from rest. Q13. The ceiling of room is 3 m bove the floor. bll of mss m kg hngs from n elstic string ttched to the ceiling. The nturl length of the string is 1.2 m nd in equilibrium the bll hngs 1.5 m below the ceiling. The bll is now pulled down to the floor with the string stretched. Find the speed with which the bll hits the ceiling when relesed. Q14. () n ellipse with semi-xes 0.8 m nd 1.2 m is cut from uniform lmin s shown (ll lengths in m). Find the position of the centre of mss of the resulting plne figure. The object is pivoted bout point nd hngs in equilibrium. If the lmin hs re mss density 5 kg m 2 how much work must be done to rotte the object so tht point is verticlly bove? Mechnics Exmples for Topic (Energy) - 3 Dvid psley

4 Q15. (Exm 2015) mss M is llowed to slide long smooth verticl pole, restrined by flexible mechnism which is pivoted t point O (see right). horizontl force P is pplied t the elbow. The mss of the mechnism my be neglected. () Write expressions for the distnces x nd y in the figure, in terms of the rm lengths nd ngle θ. Write expressions for the infinitesiml chnges δx nd δy to these lengths when there is n infinitesiml chnge δθ in the ngle. (c) Use the Principle of Virtul Work to find n expression for the force P required to hold the mechnism in the position defined by the rbitrry cute ngle θ. P x O M y k x P Q16. In the ssembly shown the links re ech of length 2, whilst the spring hs unstretched length nd stiffness k. Use the Principle of Virtul Work to find n expression for the force P in terms of the distnce x. Q17. mss M is suspended from the end of n rm of length 2 nd is lifted by the ppliction of horizontl force F to side link of length freely jointed t the centre of the min rm. If the msses of both links re negligible, use the Principle of Virtul Work to show tht, t equilibrium, M F = Mg tn θ where θ is the ngle mde by the min rm with the verticl. F P O Q Q18. (ee) The digrm shows smooth wire bent into the form of circle in verticl plne. ring P is threded on the circle nd tied to light inextensible string which psses over pulley O t the highest point of the circle. prticle of the sme mss s the ring hngs t the other end of the string. Use n energy method to find two positions of equilibrium, one stble nd the other unstble. Mechnics Exmples for Topic (Energy) - 4 Dvid psley

5 Q19. uniform rod of mss 2m nd length is smoothly hinged t. light inextensible string is ttched to its other end nd psses over smooth ril t to crry mss m. The ril t is distnce verticlly bove. The totl length of the string is l. () Find n expression for the potentil energy reltive to point in terms of the ngle θ mde by the rod with the upwrd verticl. Show tht, provided the length of the string is sufficient, there re two positions of equilibrium nd determine which is stble. m mss 2m k=3mg/ D Q20. (Exm 2014) Four uniform rods, ech of mss m nd length, re freely jointed t,,, D nd hng from s shown. Points nd re connected by spring of stiffness k = 3mg/ nd unstretched length. mkes ngle θ with the verticl. () Show tht the potentil energy of the system is given by: Show tht there re two positions of equilibrium nd determine their stbility. Q21. (Exm 2012) In the sprung system shown light pltform supporting mss of 6 kg is held in plce by two light rods, ech of length 0.4 m nd smoothly pinned t both ends. Other distnces between pins re s given in the figure. spring of stiffness k = 500 N m 1 nd unstretched length 0.2 m connects opposite hinges s shown. The msses nd cross-sectionl dimensions of the rods nd supporting pltform cn be neglected. 0.4 m 0.4 m 0.4 m 0.4 m 6 kg h () Write down expressions for the height h of the pltform bove hinge nd the extension of the spring in terms of the ngle θ shown in the figure. Write down n expression for the totl potentil energy of the system in terms of the ngle θ. (c) Find n ngle θ in the rnge 0 < θ < 180 such tht there is position of equilibrium, nd show tht this equilibrium is stble. (d) Show tht there is position of unstble equilibrium when θ = 180. Mechnics Exmples for Topic (Energy) - 5 Dvid psley

6 Q22. In the mechnism shown, mss m is free to slide on frictionless smooth verticl rod. It is connected rigidly to x second mss m by light horizontl strut. The mechnism m k is constrined by light struts of length which re D E m smoothly hinged t, nd nd hve smooth slider on the connecting rod t D. The spring shown hs stiffness k nd unstretched length equl to tht of the connecting y rod. The dimensions of the msses my be neglected. () Write expressions for the horizontl nd verticl lengths x nd y respectively, in terms of the single degree of freedom θ. Drw the ctive force digrm. (c) Determine ny possible positions of equilibrium (i.e. vlues of θ) using: (i) virtul-work (ii) potentil-energy methods, distinguishing the rnge of vlues of k for which they my occur. (You should find tht for some vlues of k there re two possible positions of equilibrium, whilst for other vlues of k there is only one). (d) (Hrd) Use the potentil-energy method to determine the stbility of ech of the positions of equilibrium in (c). D Q23. (ee) smll lmp of mss m is t the end of light rod of length 2 ttched t to verticl wll in such wy tht the rod cn rotte freely bout in verticl plne. spring D of nturl length nd modulus of elsticity λ is joined to the rod t its midpoint nd to the wll t point D distnce verticlly bove, s shown. Show tht if λ > 4mg the lmp cn hng in stble equilibrium, nd find the ngle D. Q24. rhombus D is formed of light rods, ech of length, which re pin-jointed together nd hung from point. Mss m is hung from point. Joints nd D re connected by spring of nturl length nd modulus of elsticity 2mg. Find n expression for the potentil energy of the system nd clculte the ngle θ mde by ech rod with the verticl. k = 2mg D (Note: you will need to use repeted tril, or other numericl method, to solve for θ). m Mechnics Exmples for Topic (Energy) - 6 Dvid psley

7 Q25. winch operting t 3 kw pulls box of mss 200 kg t stedy speed up slope inclined t 25 to the horizontl. Find the speed of the box if the slope is: () smooth; rough, with coefficient of friction between box nd slope of 0.3. Q26. box of mss 2 kg is plced on rough, horizontl plne. It is to be pulled long by n elstic string with nturl length 0.5 m nd modulus of elsticity 15 N. The coefficient of friction between box nd plne is 0.3. If the string is pulled horizontlly from its free end how long would the string be when the box strted to move? 30 o 20 kg 15 kg Q27. locks nd of mss 20 kg nd 15 kg respectively re connected by light inextensible string s shown. The coefficient of friction between block nd the sloping plne is 0.2. Use energy methods to determine: () which block rises; the speed of the blocks when ech hs moved 0.5 m. Q28. In bungee jump volunteer of mss 70 kg drops from bridge, tethered to his jump point by n elstic cble of unstretched length = 20 m nd elstic modulus 3000 N. Ignoring energy losses, nd ssuming he hits nothing below(!), find: () the jumper s mximum distnce of fll; his velocity when the cble first becomes tut; (c) his mximum fll velocity. If the cble were to brek t strin of 30%, find: (d) the speed with which he would hit the wter 50 m below the bridge. Q29. ctpult consists of light elstic string of elstic modulus λ = 200 N nd unstretched length 0.2 m ttched to fixed points P nd Q distnce 0.3 m prt horizontlly. mss of 0.1 kg is plced t the centre of the string nd pulled verticlly downwrd to distnce 0.2 m below the fixed points before being relesed. Neglecting friction find: () the speed with which the mss leves the string; the mximum height bove the fixed points to which it rises. P 0.3 m Q Mechnics Exmples for Topic (Energy) - 7 Dvid psley

8 Q30. (Exm 2011; extended by (e)) prticle of mss 4 kg is plced on smooth plne sloping t 30 to the horizontl s shown. The prticle is tethered to point O t the top of the slope by light string of unstretched length 3 m nd elstic modulus λ = 90 N. () lculte the extension of the string nd the distnce of the prticle from O t equilibrium. 30 o 4 kg O The prticle is pulled further down the slope so tht its distnce from O is 6 m. lculte the elstic potentil energy stored in the string t this point. (c) Show tht, if relesed from rest, it will rech the top of the slope nd find its speed when it gets there. The slope is now roughened so tht the coefficient of friction between prticle nd plne is 0.3. (d) Find the mximum friction force between prticle nd plne nd hence the mximum nd minimum distnces from O t which it cn remin t rest. (e) y considering the work done by friction, show tht, if relesed from rest t the sme position s in prt, the prticle will not rech the top of the rough slope nd determine how close to O it cn get. Q31. (Exm 2014) smll object of mss 2 kg is plced on sloping surfce mking ngle 25 to the horizontl. It is tethered to fixed point on the slope by light cord of stiffness k = 10 N m 1 nd length 4 m. The coefficient of friction between object nd slope is μ = 0.3. If it is relesed from the tethering point how fr down the slope will the object slide? 25 o 2 kg Q32. solid brss weight of mss M consists of circulr cylinder with height equl to its dimeter surmounted by hemisphericl dome of the sme dimeter. It is plced on plne sloping t 10º to the horizontl. The coefficient of friction between weight nd plne is 0.1. () Find the centre of mss of the weight nd confirm tht it will not topple over. The bse of the weight is tethered to the top of the slope by string of length nd elstic modulus λ = 4Mg. The weight is relesed from rest with the string just tut. onfirm tht the weight will slide down the slope. (c) Find (s multiple of ) the mximum distnce tht the weight will slide. Mechnics Exmples for Topic (Energy) - 8 Dvid psley

9 Q33. (Exm 2013) block of mss 2.5 kg, initilly t rest, slides distnce 5 m down rough plne before it hits spring. It then rebounds up the slope. The plne is inclined t 30 to the horizontl nd the coefficient of friction between block nd plne is 0.2. The spring hs stiffness 200 N m 1. Find: () the mgnitude of the friction force; 30 o the speed of the block when it hits the spring; (c) the mximum compression of the spring nd the elstic potentil energy stored t this point; (d) the minimum distnce from its strting point tht the block ttins on its rebound. k = 200 N/m Q34. (Exm 2018) climber of mss 75 kg is climbing verticl cliff nd is secured by sfety rope with stiffness k = 2300 N m 1. He loses his footing nd when he hs fllen 4 m the rope becomes tut. How much further does he fll before he is brought instntneously to rest? Q35. (Exm 2015; extended by prt (c)) block of mss 3 kg slides, from rest, distnce 5 m down 3 kg rough plne slope ngled t 12º to the horizontl before 5 m hitting spring of stiffness 1.8 N m 1. The coefficient of o 12 friction between block nd slope is () Find: (i) the speed with which the block hits the spring; (ii) the mximum compression of the spring. fter compressing the spring, does the block begin to rise bck up the slope? Justify your nswer. (c) Wht is the size of the friction force in prt? 1.8 N/m Q36. (Exm 2016) The 2 kg collr, of negligible dimensions, strts from rest t nd slides long fixed rod, which is curved in the shpe of qurter-circulr rc (rdius 300 mm) in the verticl plne. It is pulled by constnt horizontl force of 30 N. () Wht work is done by: 2 kg (i) the 30 N force; (ii) grvity; when the collr is moved from to? If the rod is smooth, wht is the speed of the collr when it reches? (c) If the rod is rough nd the speed t is hlf the vlue computed in prt, wht is the verge friction force on the collr? 300 mm 30 N Mechnics Exmples for Topic (Energy) - 9 Dvid psley

10 Q37. wter jet of dimeter 100 mm nd speed 15 m s 1 impcts on the buckets of Pelton wheel ( modern design of wter wheel). () If ll the kinetic energy of the wter is converted into rottionl energy of the wheel, nd thence into electricl energy, wht would be the mximum output power? Give some resons, other thn friction, why this power could not be chieved. (Tke the density of wter s 1000 kg m 3.) Jet ucket Q38. () D electric motor is used to lift 200 kg mss t stedy 0.7 m s 1 vi pulley nd counterweight of 60 kg (see the figure). If the electricl input is 1.0 kw, wht is the combined electricl nd mechnicl efficiency η of the system? If the cble from the motor to the 60 kg mss were suddenly to brek, how much higher would the 200 kg mss rise before coming momentrily to rest? 60 kg 200 kg 1 kw motor 0.7 m/s Mechnics Exmples for Topic (Energy) - 10 Dvid psley