Week 4: Friction and Tension M. [email protected] This lecture is being recorded The University of Birmingham is committed to enhancing the student learning experience. It is with this purpose in mind that the University promotes the recording of events such as lectures. This lecture will therefore be recorded and may be made available online for registered students to review and for revision purposes. The recording may be stored in the cloud and any personal information within the recording will be processed in accordance with the Data Protection Act 1998. If you ask a question or make a comment, your voice may appear on the recording. Please ask the Lecturer to pause the recording if you do not want your question or comment recorded. A substantial contribution is considered to be anything more than merely answering questions or participating in a group discussion. Where you make a substantial contribution to the delivery of recorded events, a signed Consent Form will be obtained prior to the recording being made available for viewing. The Consent Form will address your personal information and any copyright or other intellectual property in the recording. Please let your lecturer know if you do not wish to be recorded, and he or she will make alternative arrangements to enable you to contribute to any discussions and/or ask questions without being recorded Facts about friction • The maximum value of friction 퐹 is decided by the nature of the two surfaces in contact and the normal contact force 푁 between them • Friction can be calculated from the equation 퐹 = 휇푁 where 휇 is the coefficient of friction – a dimensionless number with value that depends on the nature of surfaces • 휇 is lower for surfaces sliding against each other (coefficient of sliding or kinetic friction) than it is for the same surfaces at rest (coefficient of static friction) • 휇 varies with temperature and humidity • Friction is independent of contact area (think: why do some car drivers buy wide tyres?) • The direction of friction is opposite to the direction of the motion at the point of contact Direction of friction Question: which way is friction? Question Find the relationship between the coefficient of friction 휇 and the angle 휃 if the box is sliding at constant velocity. 휃 Friction: the reality Friction: the reality https://www.youtube.com/watch?v=FO5bq6x_Tws Friction: the reality Limiting friction, angle of friction The maximum value of static friction, when motion is impending, is referred to as limiting friction. The maximum angle before which one of the items will begin sliding is called the angle of friction; it is defined as: 푡푎푛 휃 = µs where θ is the angle from vertical and µs is the static coefficient of friction between the objects. Elasticity • Elasticity is the property of an object or material which causes it to be restored to its original shape after distortion. • When stretched, elastic objects exert a restoring force which tends to bring them back to their original length. • This restoring force is generally proportional to the amount of stretch, as described by Hooke’s law: 퐹 = −푘푥 Hooke’s law Why the minus? F Gradient of graph = k Hooke’s law is valid only up to the x elastic limit. Springs in series and in parallel Find the spring constant 푘푠 of the system of springs in series in relation to the constants 푘1 and 푘2 푘1푘2 Answer: 푘푠 = 푘1+푘2 Find the spring constant 푘푝 of the system of springs in series in relation to the constants 푘1 and 푘2 Answer: 푘푝 = 푘1 + 푘2 Springs in series and in parallel Find the spring constant 푘푠 of the system of springs in series in relation to the constants 푘1 and 푘2 푘1푘2 Answer: 푘푠 = 푘1+푘2 Find the spring constant 푘푝 of the system of springs in series in relation to the constants 푘1 and 푘2 Answer: 푘푝 = 푘1 + 푘2 Springs in series and in parallel Find the spring constant 푘푠 of the system of springs in series in relation to the constants 푘1 and 푘2 푘1푘2 Answer: 푘푠 = 푘1+푘2 Find the spring constant 푘푝 of the system of springs in series in relation to the constants 푘1 and 푘2 Answer: 푘푝 = 푘1 + 푘2 Question A load is suspended from several identical springs in series. If another spring is added in the series and the load remains the same what happens to the extension of each spring? Modulus of elasticity • Elastic strings: strings which are not a fixed length. Under tension their length changes from a natural length 푙 to a new length 푙 + 푥 where 푥 is the extension of the string. • Hooke's Law: the tension in an elastic string (or spring), T, is found as: λ푥 푇 = 푙 where λ is the modulus of elasticity in Newtons. Spring constant - modulus of elasticity relationship 퐹 = 푘푥 (where 푘 is the spring constant in Nm-1) λ푥 푇 = (where λ is the modulus of elasticity in N) 푙 λ 푘= 푙 Worked example A string with modulus (of elasticity) 10 N has a natural length of 2m. What is the tension in the string when its length is 5m? • string extension = 5-2 = 3m λ푥 10 ∗3 • 푇 = = =15N 푙 2 So the tension in the string is 15N. Hooke’s law: springs in tension and compression • Hooke’s law region is a continuous straight line • The positive and negative regions are not symmetrical • The extent of the linear region is not the same in compression and extension • The curvatures of the non- linear regions are not the same Non-linear region Elastic potential energy • elastic strings or springs store energy 퐸 when Applied force extended/compressed • this energy is equal to the work done against the force exerted by the elastic string or spring by virtue of its tension extension • energy = work done = 1 1 1 λ 퐸 = 퐹푥 = 푘푥2 = 푥2 area under graph 2 2 2 푙 Question 1 A string (which may be assumed to be light), with natural length l and modulus of elasticity 4mg, is tied at one end to a fixed point A. The string passes over a smooth peg B (which is fixed) and at the other end a particle P, of mass m, is attached. The particle hangs in equilibrium. The distance between A and B is l, and AB is inclined at 60° to the vertical. a) Find, in terms of l, the length of the vertical portion BP of the string. b) Show that the magnitude of the force exerted by the string on the peg is 푚푔 3. Answer: (a) BP= l/4 Question 1 A string (which may be assumed to be light), with natural length l and modulus of elasticity 4mg, is tied at one end to a fixed point A. The string passes over a smooth peg B (which is fixed) and at the other end a particle P, of mass m, is attached. The particle hangs in equilibrium. The distance between A and B is l, and AB is inclined at 60° to the vertical. a) Find, in terms of l, the length of the vertical portion BP of the string. b) Show that the magnitude of the force exerted by the string on the peg is 푚푔 3. Answer: (a) BP= l/4 Question 2 A uniform rod( which may be regarded as thin) AB, of length 3m and mass 5kg, is freely pivoted to a fixed point A. A light elastic string BC, with modulus of elasticity, 30N, has one end C fixed to a point at the same level as A, where AC = 5m. When the system is in equilibrium, ∠퐴퐵퐶 = 90°. Calculate: a. The tension, in Newtons, in the string, b. The natural length of the string. Answer: (a) 0.3mg = 15N (b) 1.3m=1m Question 2 A uniform rod( which may be regarded as thin) AB, of length 3m and mass 5kg, is freely pivoted to a fixed point A. A light elastic string BC, with modulus of elasticity, 30N, has one end C fixed to a point at the same level as A, where AC = 5m. When the system is in equilibrium, ∠퐴퐵퐶 = 90°. Calculate: a. The tension, in Newtons, in the string, b. The natural length of the string. Answer: (a) 0.3mg = 15N (b) 2.6m=3m Homework From isaac physics: board named “week 3 Newton’s laws” From Canvas: Math centre documents • 1.8 Equilibrium of a particle • 2.1 Newton’s first law of motion • 2.2 Newton’s second law of motion • 2.3 Types of force • 2.4 Parallel forces acting together • 2.5 Motion due to gravity • 2.6 Forces acting at an angle: Resolving forces • 2.7 Particles on a slope • 2.11 Newton’s third law of motion From the web: Conceptual physics videos 21, 22, 23, 39, 40, 41, 42 • http://www.conceptualphysics.com/21-30.html • http://www.conceptualphysics.com/31-40.html • http://www.conceptualphysics.com/41-50.html.