1. A piston of mass 4.53kg is travelling in a tube with a velocity of 15.24m/s and engages a spring and a damper, as shown in the figure.Determine the maximum displacement of the piston after engaging the spring-damper. How many seconds does it take? (12 points)

3. Do Problem 8.131 from the textbook.[Statement: Water flows through the Venturi meter shown in Figure 8.131. The specific gravity of the manometer fluid is 1.52. Determine the flow rate.]

1. A system has a transfer function G(s) given by: G(s)=\frac{5}{(2 s+3)(s+4)} Showing all your working, determine the response of the system in the time domain to a step input at time t = 0 of magnitude 1. Hence state whether the system is stable or unstable.

5.) Derive the differential equation for a motor driving a load through a gear system as shown in Figure 17.27 on page 438, which relates the angular displacement of the load with time.

9.11 Figure P9.11 shows a 1-DOF mechanical system driven by the displacement of the left end, (), which could be supplied by a rotating cam and follower (see Problem 2.2). When displacements X() = 0 and r = 0 the spring k is neither compressed nor stretched. The system purameters are Im = 2 kg. k = 50K) N/m,and h = 20N-s/m. Determine the frequency response if the position input is x(1) = 0.04 sin 50r m.

The thin homogeneous 300 lb plate is hanging from a cable attached to point O when it is subjected to an impulse of -20k lb.s at the corner A. Determine the angular velocity vector of the plate immediately after the impulse occurs.

10. Two common methods of improving fuel efficiency of a vehicle are to reduce the drag coefficient and the frontal area of the vehicle. Consider a car with 1.85m width and 1.75 m height, with a drag coefficient of 0.30. Determine the amount of fuel and money saved per year as a result of reducing the car height to1.50 m while keeping its width the same. Assume the car is driven 25,000 km(15,000 miles) a year at an average speed of 100 km/h. Take the density and price of gasoline to be 0.74 kg/L and $1.04/L. Also assume the density of air to be1.20 kg/m, the heating value of gasoline to be 44,000 kJ/kg, and the overall efficiency of the car's drive train to be 30%. (Hint: Lecture Notes)

Problem 2. (a) A tachometer has an analog display dial graduated in 5 rpm increments.The user manual states an accuracy of 1% of reading. Estimate the uncertainty in the reading at 10 rpm, 500 rpm and 5000 rpm. (b) A certain obstruction type flow meter (orifice, venturi, nozzle), shown in the following figure is used to measure the mass flow rate of air at low velocities. The relationship describing the flow rate is: \dot{m}=C A \sqrt{\left[\frac{2 g_{c} p_{1}}{R T_{1}}\left(p_{1}-p_{2}\right)\right]} where, * C = empirical-discharge coefficient. * A = flow area * T1 = upstream temperature * R = gas constant for air * Pi and p2 = upstream and downstream pressures, respectively. Calculate the percent uncertainty in the mass flow rate for the following conditions:

3. Consider a system that has a transfer function Find the response x(t) when the input f(t) is \frac{X(s)}{F(s)}=\frac{8}{s+2} a unit step function u(t)=1 for t >= 0, and an impulse (Dirac delta) function 8(t) occurring at time t = 0.

5.1Derive the state-variable equations for the system that is modeled by the following ODES where a, w, and z are the dynamic variables and v is the input. 0.4 \dot{\alpha}-3 w+\alpha=0 0.25 z+4 z-0.5 z w=0 \ddot{w}+6 w+0.3 w^{3}-2 \alpha=8 v