11exam3solotion - Exam 3 NAME z O y x F 1 F 2 F 3 x,y,z P 1...

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Unformatted text preview: Exam 3 NAME: z O y x F 1 F 2 F 3 ( x,y ) ,z P 1. A particle P of mass m is released from rest at the initial position P ( x ,y ,z ). The particle is acted upon by the force system F 1 = mk 1 ı , F 2 = mk 2 , and F 3 = mk 3 k . The effect of gravity on the particle is ne- glected. For numerical application use x = 1 m, y = 2 m, z = 3 m, k 1 = 10 m/s 2 , k 2 =- 10 m/s 2 , k 3 = 10 m/s 2 . a) Develop the differential equation or equations describing the motion of the particle. This equation or these equations should involve the coordinates x , y , z of the particle, various time derivatives of these coordinates, and the pa- rameters m , k 1 , k 2 , and k 3 . b) Find the acceleration of the particle; c) Determine the distance the particle is from the origin a time t = 1 s after it has been released. y x O R θ v P P 1 P ( m ) R 2. A particle P of mass m is propelled inside a circular track of radius R as shown in the figure. At the initial time t = 0 the particle is at the point P ( R, 0) and has the initial conditions θ (0) = 0 and ˙ θ (0) = 1 rad/s. For numerical application use R = 1 m, m = 1 kg, g = 10 m/s 2 . a) Develop the differential equation or equations describing the motion of the particle. This equation or these equations should involve the angle θ , various time derivatives of that angle, and the parameters m , R , and g ; b) Determine the acceleration of the particle at t = 0. c) Find the expression of the normal contact force N between the circular track and the particle at t = 0. O y x P ( m ) B h α A 3. A particle P of mass m is traveling down an inclined surface as shown in the figure. The particle P is released from rest at the point A . The angle between the inclined surface and the horizontal is α and the point A is located at the vertical distance h . The coefficient of friction between the particle and the surface is μ . For numerical application use m = 1 kg, α = 45 ◦ , h = 1 m, μ = 0 . 1, and g = 10 m/s 2 . a) Find the work done by the particle from A to B ; b) Find the velocity of the particle at point B where the inclined surface intersects the horizontal axis. 1. The equations of motion of the particle are m ¨ x = mk 1 , m ¨ y = mk 2 , m ¨ z = mk 3 . or ¨ x = k 1 , ¨ y = k 2 , ¨ z = k 3 ....
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11exam3solotion - Exam 3 NAME z O y x F 1 F 2 F 3 x,y,z P 1...

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