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Elasticity Hooke's Law : the extension in an elastic string is proportional to the applied force . T= l x l x = extension l = natural length = modulus of elasticity and depends on the material involved NOT the length. If the string is doubled in length then the extension will equal the original length l So T= ll l = = force required to double the length of string Elastic Strings / Springs In equilibrium T = mg l x = mg l x = m gl l T= l x mg l The equilibrium extension is often called e e = m gl l Ex1 A 2kg ball is attached to a 3m string of =50N. The string is attached to a point A. The ball hangs in equilibrium. Find the length of the string. A T In equilibrium 50e T = l e= = 2g l 3 e = 1.2 So length string = 4.2m in equilibrium 2g A mass of 2kg is hung from a 50cm string with = 30N. It is pulled down 10cm from the equilibrium position find its initial acceleration. T In equilibrium T – 2g = 0 Solve for e e= T = l e = 2g so 2g ´ 0.5 30 l = 0.33m e = equilibrium extension 2g When it is pulled down 10 cm it is no longer in equilibrium so use resultant force = ma T – 2g = 2a but T = l x 30 ´ 0.43 - 2g = 2a 0.5 l Solving So a = 2.9ms–2 l x – 2g = 2a l T= Springs The formula also applies if a spring is either stretched or compressed. Ex.1 An elastic spring of modulus of elasticity 30N and natural length 20cm is compressed in a vice to a length of 17cm. Find the force exerted by the jaws of the vice. The compression in the spring is the same throughout and is equal to the force exerted by the jaws of the vice Using T = l x l T = 30 ´ 0.03 0.2 = 4.5N Ex.2 An 18cm deep mattress containing 14 springs is compressed by 0.7cm when a 75kg sleeper lies on it. Find the force to compress it 1cm If a force of 75g is applied to the mattress then each spring experiences a compression force of: F = 75g = 52.5N 14 Using T = l x l 52.5 = l ´ 0.007 0.18 = 1350 Energy stored in a stretched elastic string If a constant force moves a distance x then the work done = F s F s W=Fs i.e. area under the line Stretching a string needs an increasing force. F W or k = a r ea u n d er lin e s = ò F ds = ò T ds = ò λs λs 2 ds = l So the work done stretching a string a distance x is : W = l x 2l 2l 2 But in stretching a string then energy is acquired so: The Elastic energy acquired = the work done stretching the string The Elastic energy is called Elastic Potential energy E.P.E 2 l x E.P.E = 2l So now the total energy equation is : Total mechanical Energy = K.E + P.E + E.P.E A 2kg ball is attached to a 3m string of =50N. The string is attached to a point A. The ball hangs in equilibrium. Find the length of the string. In equilibrium 50e T = l e= = 2g l 3 e = 1.2 A T 2g So length string = 4.2m in equilibrium The ball is now lifted up to the point A and dropped. Find the velocity at the equilibrium point and the max length of string. A From A to N the body accelerates as there is no upward force as it falls. N = natural length = 3m A From A to N the body accelerates as there is no upward force as it falls. From N to E the body continues to accelerate as the downward force is greater than the upward force. The body has max velocity at E as at this point the upward and downward forces balance N = natural length = 3m E = equilibrium length = 4.2m A From A to N the body accelerates as there is no upward force as it falls. From N to E the body continues to accelerate as the downward force is greater than the upward force. The body has max velocity at E as at this point the upward and downward forces balance N = natural length = 3m E = equilibrium length = 4.2m B From E to B the body decelerates as the upward force is now greater than the downward force. It stops at B. A At A PE = 2g4.2 KE = 0 EPE = 0 Make the equilibrium point have zero P.E. So points below this point have negative P.E. N E B A At A PE = 2g4.2 KE = 0 EPE = 0 Make the equilibrium point have zero P.E. So points below this point have negative P.E. N 1.2m E PE = 0 KE = 2v2 EPE h B PE = -2gh KE = 0 EPE = 50 ´ 1.2 2 = 2´ 3 50( h + 1.2) 2 2´ 3 A At A PE = 2g4.2 KE = 0 EPE = 0 Energy at A = Energy at E 2g4.2 = 0 + 2v2 + 50 ´ 1.2 2 2´ 3 Solve for v g = 10ms-2 0 = v2 – 72 v = 8.5ms-1 N E PE = 0 KE = 2v2 EPE = h B PE = -2gh KE = 0 EPE = 50 ´ 1.2 2 2´ 3 50( h + 1.2) 2 2´ 3 A At A PE = 2g4.2 KE = 0 EPE = 0 Energy at A = Energy at B 2g4.2 = -2gh 50( h + 1.2) 2 + 2´ 3 0 = 50 h2 – 72 6 h = 2.93m Solve quadratic for h So total distance = 7.13m (add on 4.2) N h E B PE = 0 PE = -2gh KE = 2v2 KE = 0 EPE = EPE 50 ´ 1.2 2 2´ 3 50( h + 1.2) 2 = 2´ 3 Ex 2 Mass 2kg hung from a 1m spring with = 50N. Find the extension. It is then pulled down a further 20cm. Find initial acceleration, velocity at equilibrium point and max height. A T In equilibrium 50e T = l e= = 2g l 1 e = 0.4m So length string = 1.4m in equilibrium 2g A At A PE = 2g1.4 KE = 0 EPE = 0 initial acc. T – 2g = 2a subst x = 0.6 and = 50 50 ´ 0.6 - 2g = 2a 1 a = (500.6 - 20) = 5ms-2 0.4 N E PE = 0 KE = 2v2 EPE = 50 ´ 0.4 2 2´ 1 EPE = 5 0( 0 .6 ) 2 2´ 1 0.2 B PE = -2g 0.2 KE = 0 A At A PE = 2g1.4 KE = 0 EPE = 0 Energy at B = Energy at E -2g(0.6) + 1 = v2 v = 1ms-1 0.4 50 ´ 0.6 2 2´ 1 = 0 + 2v2 + 50 ´ 0.4 2 2´ 1 So velocity at E = 1ms-1 N E PE = 0 KE = 2v2 EPE = 0.2 B PE = -2g(0.6) KE = 0 50 ´ 0.4 2 2´ 1 EPE = 5 0( 0 .6 ) 2 2´ 1 A At A PE = 2g1.4 KE = 0 EPE = 0 Energy at B = Energy at N -2g(0.2) + 50 ´ 0.6 2 2´ 1 = 2g0.4 + 2v2 v =-3 impossible so it does not reach N 0.4 N = 1m E = 1.4m PE = 2g0.4 KE = 2v2 EPE = 0 PE = 0 KE = 2v2 EPE 50 ´ 0.4 2 = 2´ 1 0.2 B = 1.6 PE = -2g0.2 KE = 0 EPE = 5 0( 0 .6 ) 2 2´ 1 Finding the maximum height – M A At A PE = 2g1.4 KE = 0 EPE = 0 Energy at B = Energy at M -2g(0.2) + 50 ´ 0.6 2 2´ 1 = 2g(0.4- x) + 50 ´ x 2 2´ 1 x = 0.2m or 0.6m But 0.6m is at the bottom so x = 0.2m x 0.2 N M E B at M PE = 2g(0.4-x) KE = 0 EPE = PE = -2g(0.2) KE = 0 EPE = 50 ´ x 2 2´ 1 5 0( 0 .6 ) 2 2´ 1 So the length of the spring at the max height is 1.2m i.e 0.2m above E Finding the maximum height – M A At A PE = 2g1.4 KE = 0 EPE = 0 Energy at B = Energy at M -2g(0.2) + 50 ´ 0.6 2 2´ 1 = 2g(0.4-x) + 502 ´´ x2 1 x = 0.2m or 0.6m But 0.6m is at the bottom so x = 0.2m x N M 0.2 B So the length of the spring at the max height is 1.2m i.e 0.2m above E