12. In an RLC series circuits, \(v_R(t)\), \(v_L(t)\), \(v_C(t)\) are the potential difference across a resistor and an inductor. What are the relation of \(v_R(t)\) in relation of \(i(t)\) and \(v_L(t)\) with \(i(t)\)?      (13)      (Hint: in phase; leads by 90°; lags by 90°)

1. A ball is thrown horizontally from the height of 20 m and hits the ground with a speed that is three times its initial speed. The initial speed of the ball is      (1)      m/s.

2. A 1000 kg boat is traveling at 60 km/h when its engine is shut off. The magnitude of the frictional force \(f_k\) between the boat and water is proportional to the speed \(v\) of the boat: \(f_k = 7v\), where \(v\) is in meters per second and \(f_k\) is in newtons. The time required for the boat to slow down to 30 km/h is      (2)      seconds.

3. A 4.0 kg bundle starts up a 30° incline with 128 J of kinetic energy. How far will it slide up the incline if the coefficient of kinetic friction between the bundle and the incline is 0.30?      (3)      m.

4. A block of mass \(m_1 = 1.60\) kg is moving to the right with a speed of 4.00 m/s on a frictionless horizontal track, collides with a mass-less spring attached to a second block of mass \(m_2 = 2.10\) kg, moving to the left with a speed of 2.50 m/s on the same track. Determine the maximum distance by which the spring is compressed during the collision.      (4)      m. (The spring has a spring constant of 600 N/m)

5. A uniform ball, of mass \(M = 5.00\) kg and radius \(R\), rolls smoothly from rest down a ramp at angle \(\theta = 30.0°\). The ball descends a vertical height \(h = 1.20\) m to reach the bottom of the ramp. The speed of the ball at the bottom is      (5)      m/s.

(6)

(7)

7. You are trying to raise a bicycle wheel of mass \(m\) and radius \(R\) up over a curb of height \(h\). To do this, you apply a horizontal force \(F\). What is the smallest magnitude of the force \(F\) that will succeed in raising the wheel onto the curb when the force is applied to the center of the wheel?      (8)     

8. What are the electric fields of an isolated sphere of radius \(R\), with uniform volume charge density \(\rho\), at a point outside the sphere?      (9)     

9. An air-filled parallel plate capacitor is connected to a battery of voltage \(V_0\). There is charge \(Q_0\) on each plate, energy \(U_0\) stored between the plates and an electric field of \(E_0\) between the plates. The capacitor remains connected to the battery and a dielectric material with a dielectric constant of \(K\) is then placed between the plates of the capacitor. What are the new values of \(V\), \(Q\), \(U\), and \(E\)?      (10)     

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