ENGR 2120 Dynamics Exam 03 (6:30‐9:00 pm) at WJOE@tnstate.edu Note) Follow the homework format. 1. (30 points) Figure 01. A 10 kg mass with a spring constant 40 N/m is shown in the figure 1. 1) Find the acceleration as a function a distance. 2) Find the velocity of the mass at 2 sec. 3) Find the velocity of the mass at 0.2 m. 4) Find the maximum distance using the newton’s law. 5) Find the velocity of the mass at 0.3m using the work‐KE method. 2. (20 points) 1) Find the homogenous solution of the system in the figure 1. 2) Find the particular solution of the system in the figure 1. 3) Find the general solution of the system in the figure 1 when x(0)=0, v(0)=1m/s. 4) Design a new k (spring constant) to oscillate the system with frequency of 1 Hz in the figure 1. 3. (20 points) The mass of hammer is 10 kg, spring constant k=80 N/m, and the rail is frictionless. The mass with two springs at rest start moves from pos. 1 to pos. 2 to hit workpieces as shown in fig. 2. 1) Find the velocity of hammer at pos. 2 using Newton’s Law. 2) Find the velocity of hammer at pos. 2 using the work‐KE method. 3) Find the velocity of hammer at pos. 2 using the conservation of energy. Figure 2. 4. (30 points) Where m=5 kg, k=125 N/m (a) Find the governing ODE equation. (b) Determine the frequency of the motion in Hz (c) Find the displacement x(t) when t=0.4 sec with initial conditions x(t=0)= 0 m and v(t=0)= 1 m/s (d) In order to have the frequency of motion 2 Hz, find a new kEquation sheet When a is zero, v is constant. ݐݒ ݏൌݏ When a is constant, ଵ ݐݒ ݏൌݏ ଶ ݒ ଶ ൌݒ ,ݐܽ ݒൌݒ ,ଶݐܽ ଶ 2ܽሺݏ െ ݏሻ, Other definition ܽ ൌ ௗ௩ ௗ௧, ݒ ൌ ௗ௦ ௗ௧, ܽൌݒ ௗ௩ ௗ௦ Velocity and acceleration in tangential normal coordinates ݒൌݒ௧݁௧, ܽൌܽ௧݁௧ ܽ݁, ൌ ܽ௧݁௧ ݒ௧߱݁ Where ݒ௧ ൌ ܴ߱, ܽ௧ ൌ ௗ௩ ௗ௧ , ߱ ൌ ௗఏ ௗ௧ Position, velocity and Acceleration in polar coordinates ݎ̅ൌ ݎ݁, ݒൌݒ݁ ݒఏ݁ఏ, ൌ ݒ݁௧ ݎ߱݁ఏ, ܽ ൌ ሺܽ െ ݎ߱ଶሻ݁ ሺ2ݒ߱ ߙݎሻ݁ఏ Where ܽ ൌ ௗ௩ ௗ௧ , ݒ ൌ ௗ ௗ௧, ߱ ൌ ௗఏ ௗ௧ , ߙ ൌ ௗఠ ௗ௧ Friction force ݂ ൌ ߤܰ where N is normal force. Forces in Cartesian coordinates ܨത ൌ ܨ௫݅ܨ௬݆ ൌ ݉ሺܽ௫݅ܽ௬݆ሻ Forces in tangential‐normal coordinates ܨത ൌ ܨ௧݁ഥ ܨ ௧ ݁തത ത ൌ ݉ሺܽ௧݁ഥ ܽ ௧ ݁തത തሻ ൌ ݉ሺ ௧ݒ݀ ݀ݐ ݁ഥ ݒ߱݁ ௧ തത തሻ Forces in Polar coordinates ܨത ൌ ܨ݁ഥ ܨ ఏത݁ത ఏത ൌ ݉ሺܽ݁ഥ ܽ ఏത݁ത ఏതሻ ൌ ݉ሾ൬ ݒ݀ ݀ݐ െ ݎ߱ଶ൰ ݁ഥ ሺ2ݒ߱ ߙݎሻത݁ത ఏതሿ Spring force ܨ௦ ൌ ݇∆݇ Work‐KE method ܧܭ∆ ൌ ݔ்݀ܨ න Momentum method ݒ∆݉ ൌ ݐ்݀ܨ ൌ න ܫ Average impulsive force ܨ௩ ൌ 1 ݐଶ െ ݐଵ ݐ்݀ܨ න Conservation of momentum ݉ଵݒଵ ݉ଶݒଶ ൌ ݉ଵݒଵ′݉ଶݒଶ′ Conservation of energy ܧܭଵ ܲܧଵ ൌ ܧܭଶ ܲܧଶ Unit conversion 1 mile = 1.6 km = 1600 m = 5280 ft 1 h = 60 min = 3600 sec Restitution ݁ ൌ ݒଶ′െݒଵ′ ݒଵ െ ݒଶ