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Physics of Motion Lecturer: Mauro Ferreira Text book: University Physics (extended version with modern Physics) by Young and Freedman, Addison-Wesley, 9 th edition.

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Before we start, please remember to switch off your mobile phone

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Objectives: By the end of this lecture you should be able to: Identify centripetal forces in circular motion Include air resistance in the motion of an object Solve problems about the topics above

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Outline 2 nd Newton’s law for curvilinear motion + problems Let’s include air resistance

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X Y Uniform circular motion is the motion of a particle that moves in a circle with constant speed. For instance, a car rounding a curve with constant radius, a satellite in a circular orbit or a skater moving in a circle, all with constant speed. The so-called centripetal acceleration is perpendicular to the velocity and always points towards the centre of the circular trajectory.

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X Y Uniform circular motion is the motion of a particle that moves in a circle with contant speed. For instance, a car rounding a curve with constant radius, a satellite in a circular orbit or a skater moving in a circle, all with constant speed. The so-called centripetal acceleration is perpendicular to the velocity and always points towards the centre of the circular trajectory.

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X Y Let’s select a pair of points in the circle for which their position vectors differ by an angle . Lecture 5

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Let’s select a pair of points in the circle for which their position vectors differ by an angle . Since r 1 =r 2 =R and V 1 =V 2 =V, the corresponding velocities form an equivalent triangle. From basic geometry, we have that In the limit when t 0 Lecture 5

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X Y Therefore, the centripetal acceleration is inversely proportional to the circular radius and directly proportional to the velocity squared. Lecture 5

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But according to 2 nd law, acceleration is the result of a net force Centripetal acceleration Centripetal force

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Consider a car of mass M moving at speed V along a circular path of radius R. Identify the forces on the vehicle. R Front view

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The magnitude of the forces are: Assuming that the (static) friction coefficient is , what is the maximum speed the car can have to complete the curve in safety ? V max is mass independent !!!

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Can you estimate the maximum safe speed of a car driving on a road ? Estimates:

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Banked curves are built to reduce the risk of cars skidding off the road. Let’s see why:

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The normal component N x acts as the centripetal force Cars can round a curve with no friction

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A sphere of mass 0.5kg is attached to the end of a 1.5m-long cord. The sphere is whirled in a horizontal circle, as shown in the figure. Assuming that the cord can withstand a maximum tension of 50N, what is the maximum speed at which the sphere can be whirled before the cord breaks ?

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A small sphere of mass m is attached to the end of a cord of length R and set in motion in a vertical circle, as shown in the figure. Determine the tension in the cord when the speed of the particle is V and the cord makes an angle with the vertical.

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What if the sphere is in the top part of the circular trajectory ? In this case the tension is given by Note that the tension vanishes for. Discuss what happens for V

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A child being pushed on a swing reaches increasingly larger heights. If the swing is not able to support the load, where will it break ? T is maximum when =0.

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Conical pendulum: A small object of mass m suspended from a string of length L revolves with constant angular speed forming an angle with the vertical direction, as shown in the figure. Find an expression for the angular speed

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A single bead can slide with negligible friction on a wire that is bent into a circular loop of radius R, as shown in the figure. The circle is in a vertical position and rotates steadily around its vertical diameter with angular speed . Determine the angle .

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Is air resistance always negligible ?

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We have avoided the effect of air resistance so far, but what happens when it cannot be neglected? For objects moving in a fluid, a resistance force arises opposing the motion. This force depends on the velocity and is generally given by: Interpret the equation with you own words For a body of mass m falling vertically, we have It is instructive to see what happens when k 0. Terminal velocity is V T =mg/k.

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The physical interpretation of the VxT graph is very simple. As the object starts moving, the velocity increases and so does the air resistance up to the point where R a is as large as the weight. At this point the velocity no longer changes reaching its terminal value. We see that the air resistance is not proportional to the mass of the object. That’s why heavier objects tend to fall faster.

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What about the displacement ? Once again, it is instructive to see what happens when k 0.

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Objectives: By the end of this lecture you should be able to: Identify centripetal forces in circular motion Include air resistance in the motion of an object Solve problems about the topics above

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Puzzles & challenges E-mail: ferreirm@tcd.ie

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R = ? A projectile launched with speed V 0 at an angle with the horizontal lands on an incline of angle (see figure). Find the distance R in terms of the relevant quantities. What is the optimum angle opt for which the distance R is maximum ?

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Assignments & tutorials Tutorial sheet to be handed in on: 17/11/2003 before 12:00pm

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