Integrate this equation using the initial condition that v = V₁, at t = 0 and noting that[*--8²² [aIn () = -( (B²l²¹/Rm)Define the constant T =(1)I²R =v = V₂6Use I =Finalize This expression for v indicates that the velocity of the bar decreasesBlvR|vel-t(b) Show that the same result is found by using an energy approach.mRB²²B²6²,²RdvV-t/t)Rearrange terms:SOLUTIONCategorize The text of this part of the problem tells us to use an energy approach for the same situation. We model the entire circuit in the figure as ---Select---Analyze Consider the sliding bar as one system component possessing kinetic energy, which decreases because energy is transferring out of the bar by electrical transmission through the rails. The resistor is another system component possessing internal energy,which rises because energy is transferring into the resistor. Because energy is not leaving the system, the rate of energy transfer out of the bar ---Select--- ✓the rate of energy transfer into the resistor.(Use the following as necessary: m, v, B, R, and l.)Equate the power entering the resistor to that leaving the bar:Presistor = -PbarSubstitute for the electrical power delivered to the resistor and the time rate of change of kinetic energy for the bar:-=-1and solve for the velocity:for the current and carry out the derivative:X(1/2)mv²-B²²₂²|R( - Bl/R)₂²XXdvdtB²₂²mRJatis a constant:with time under the action of the magnetic force as expected from our conceptualization of the problem.system. Magnetic Force Acting on a Sliding BarThe conducting bar illustrated in the figure moves on two frictionless, parallel rails in the presence of a uniform magnetic field directed into the page. The bar has mass m, and its length is . The bar is given an initial velocity v; to the right and is released at t = 0.A conducting bar of length sliding on twofixed conducting rails is given an initialvelocity, to the right.SOLUTIONXXXSubstitute I =Fx = ma →xxXBlvRdvm = -dtXXBIL*XxXF* X-v/mxIXx(a) Using Newton's laws, find the velocity of the bar as a function of time.|X¹xConceptualize As the bar slides to the right in the figure, a counterclockwise current is established in the circuit consisting of the bar, the rails, and the resistor. The upward current in the bar results in a magnetic force ---Select--- ✓ on the bar as shown in thefigure. Therefore, the bar must slow down, so our mathematical solution should demonstrate that.Categorize The text already categorizes this problem as one that uses Newton's laws. We model the bar as a particle under a net forceAnalyze From the equation F = I L x B, the magnetic force is FB = -IB, where the negative sign indicates that the force is to the left(Use the following as necessary: B, m, R, v¡, t, I, ¤, and T.)Using the particle under a net force model, apply Newton's second law to the bar in the horizontal direction:x BinX*XXdvdt= m-. The magnetic force is the only horizontal force acting on the bar.

Magnetic Force Acting on a Sliding Bar The conducting bar illustrated in the figure moves on two frictionless, parallel rails in the presence of a uniform magnetic field directed into the page. The bar has mass m, and its length is f. The bar is given an initial velocity v, to the right and is released at t = 0. A conducting bar of length sliding on two fixed conducting rails is given an initial velocity, to the right. x x Substitute I = x S x x x Blv R x dv M-=- m dt x x x x x x > xx x F. x B x x -v/m x x x Bin Ⓒ (a) Using Newton's laws, find the velocity of the bar as a function of time. x x SOLUTION Conceptualize As the bar slides to the right in the figure, a counterclockwise current is established in the circuit consisting of the bar, the rails, and the resistor. The upward current in the bar results in a magnetic force ---Select--on the bar as shown in the figure. Therefore, the bar must slow down, so our mathematical solution should demonstrate that. Categorize The text already categorizes this problem as one that uses Newton's laws. We model the bar as a particle under a net force Analyze From the equation Fg = IL x B, the magnetic force is Fg = -1B, where the negative sign indicates that the force is to the left ✓✓ 1 (Use the following as necessary: B, m, R, V₁, t, I, I, and T.) Using the particle under a net force model, apply Newton's second law to the bar in the horizontal direction: F=ma → BIL x x x x X X x dv dt =m- . The magnetic force is the only horizontal force acting on the bar.

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...