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The three-dimensional motion of a particle is defined by the position
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Vector Mechanics for Engineers: Dynamics
- 5. The motion of a vibrating particle is defined by the position vector r=10(1 – e 3)î + (4e "sin15t)j, where r and t are in mm and s, respectively. 3 е 2 a. What is the total velocity vector when t = 0.5 s? 1 b. What is the total acceleration vector when t=0.5 s? To x -1 -2 Earrow_forwardAt time t = 0, the position vector of a particle moving in the x-y plane is r = 5.28i m. By time t = 0.022 s, its position vector has become (5.39i+ 0.48j) m. Determine the magnitude Vay of its average velocity during this interval and the angle 8 made by the average velocity with the positive x-axis. Answers: Vav 0 = = 20 i m/sarrow_forward1. For a particle in space, the velocity may be defined by the function, (t) = 0.8ti + 12√ej + 5k. Find the velocity and acceleration of the particle at t = 9 seconds and correctly present each as cartesian vectors.arrow_forward
- At time t = 0, the position vector of a particle moving in the x-y plane is r = 4.78i m. By time t = 0.019 s, its position vector has become (4.96i + 0.52j) m. Determine the magnitude Vay of its average velocity during this interval and the angle e made by the average velocity with the positive x- axis. Answers: Vav Ꮎ = = i i m/sarrow_forwardAt time t=0, the position vector of a particle moving in the x-y plane is r = 4.48i m. By time t = 0.030 s, its position vector has become (4.711 +0.47)) m. Determine the magnitude vay of its average velocity during this interval and the angle 9 made by the average velocity with the positive x-axis. Answers: i m/s 4arrow_forwardA revolving slotted arm OA moves a bearing P in a fixed curve shape: r(θ) = Cθ. r = radial distance to O θ = angle the arm OA makes in the x-direction C = known constant Arm OA begins from rest when θ = π/4 and rotates counterclockwise with constant angular acceleration d2θ/dt2 = α. Part 1: Calculate v(t) (the velocity vector) of the P bearing as a function of t. Show result with respect to C, α, and the unit vectors ur (r-direction) and uθ (θ-direction). Part 2: Calculate a(t) (acceleration vector) of the P bearing as a function of t. Show result with respect to C, α, and the unit vectors ur (r-direction) and uθ (θ-direction). Part 3: Calculate magnitude v of the velocity and the magnitude a of the acceleration of P when the angle of the slotted arm is θ = 3π/4.arrow_forward
- 5 (a)The x-coordinate of a particle in curvilinear motion is given by x = 3.7t3 - 4.5t where x is in feet and t is in seconds. The y-component of acceleration in feet per second squared is given by ay = 1.7t. If the particle has y-components y = 0 and vy = 3.4 ft/sec when t = 0, find the magnitudes of the velocity v and acceleration a when t = 5.9 sec. Sketch the path for the first 5.9 seconds of motion, and show the velocity and acceleration vectors for t = 5.9 sec.arrow_forwardA motorcycle cage has a curved path in the x-z plane as shown. The cage path has different curved profile, and between point o and A the path is best described as z = f(x) = ax², where a = 0.09 and b = 1.23. The motorcyclist at point M is moving at a velocity of 12.6 m/s which is increasing at a rate of 1.9 m/s?. If the horizontal position of the motorcyclist at point M is xM = 11.9 m, determine (i) the direction of the motorcyclisť's velocity. (ii) the radius of the curvature of the cage's path at point M, (iii) the normal acceleration at point M, (iv) the magnitude and direction of the motorcyclist's acceleration at point M. A (z = axarrow_forwardAn automobile P is traveling along a circular track of radius R=958.4 m. At position "A" on the track, the automobile has a speed of UA = 10.3 m/s. At this position, the driver of the automobile applies the brakes causing the speed of the automobile to change with distance s traveled along the track according to the following equation: U(S) = VA COS(0.001s) m/s (cos is in radians), where s is given in meters. Determine the magnitude of the acceleration for the driver when the automobile reaches position "B" on the track where "B" is a quarter of the distance around the track from position "A". R B O circular trackarrow_forward
- I Suppose an autonomous surface vessel (ASV) traveling with velocity TvG/O= vi₁ begins to make a turn by adjusting the thrust of its left and right thrusters, TA and TB, respectively. The center of mass of the ASV is located at G and the ASV is symmetric about its vertical axis. The ASV also experiences a drag force that is proportional to its speed and opposes its velocity. At the instant shown, the drag force is D = -kvi₁ where k is a drag coefficient. 1. To model the mass moment of inertia, approximate the ASV as consisting of three rigid bodies: a flat plate as a center body of mass 6m and two slender rods housing the propulsion assemblies, each of mass m, at the outboard sides of the vehicle. Determine the mass moment of inertia, IG, about the vertical axis passing through the center of mass G. (Hint: Use the parallel axis theorem.) 2. At the instant shown, determine the inertial acceleration vector ac/o = axi₁ + ayi2 of the center of mass and the angular acceleration a of the…arrow_forwardThe blue curve shown in Figure below display the motion of a particle in x direction with respect to time. For each of the marked points in Figure, find (a) the velocity and (b) the acceleration vector for the particle. а. 4.5 4 A 3.5 3 2.5 1.5 0.5 1 1.5 2 2.5 3.5 4 t b. Suppose that a traction on the foot maintained shown in the figure below. Draw the free body diagram and find the force F applied by the foot muscle. 35 35 20 Narrow_forward6(a) The y-coordinate of a particle in curvilinear motion is given by y = 4.5t3 - 5.9t, where y is in inches and t is in seconds. Also, the particle has an acceleration in the x-direction given by ax = 9.6t in./sec2. If the velocity of the particle in the x-direction is 12.9 in./sec when t = 0, calculate the magnitudes of the velocity v and acceleration a of the particle when t = 1.4 sec. Construct v and a in your solution.arrow_forward
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