3. A single closed loop of chain of mass m and length l rests on the surface of a smooth, frictionless cone. The chain lies in a horizontal plane. The half-angle of the cone is α. What's the tension in the chain?
(A)\( T = \frac{mg}{\tan\alpha} \)
(B)\( T = \frac{mg}{2\pi} \)
(C)\( T = mg \)
(D)\( T = \frac{mg}{2\pi\cot\alpha} \)

(E)\( T = \frac{mg}{2\pi\tan\alpha} \)

1. A particle moves along the y axis. Its position is given by the equation \( y(t) = 2 + 3t - 4t^2 \), with y in meters and t in seconds. What's its position when it changes direction?(A)2.56 m(B)1.28 m(C)4.23 m(D)2.12 m(E)3.84 m

2. A particle moves along a straight line starting at t = 0. Its position is given by the equation \( x(t) = 75t - 1.0t^3 \), where t is in seconds. When it stops (v=0), its acceleration is:(A)0(B)-73 m/s²(C)-30 m/s²(D)-9.8 m/s²(E)49 m/s²

4. If the net work done by external forces on a particle is zero, which of the following statements about the particle must be true?(A)Its velocity is zero.(B)Its velocity is decreased.(C)Its velocity is unchanged.(D)Its speed is unchanged.(E)More information is needed.

5. A car of mass m moving at a speed \( v_1 \) collides and couples with the back of a truck of mass 2m moving initially in the same direction as the car at a lower speed \( v_2 \). What is the speed \( v_f \) of the two vehicles immediately after the collision?(A)\( v_f = \frac{2}{3}(v_1 - 2v_2) \)(B)\( v_f = \frac{2}{3}(v_1 + v_2) \)(C)\( v_f = \frac{1}{3}(v_1 - 2v_2) \)(D)\( v_f = \frac{1}{3}(v_1 + 2v_2) \)(E)\( v_f = \frac{1}{2}(v_1 + v_2) \)

6. Objects A and B interact with each other via both conservative and nonconservative forces. Let \( K_A \) and \( K_B \) be the kinetic energies, U be the potential energy, and \( E_{int} \) be the internal energy. If no external agent does work on the objects then:(A)\( K_A + U \) is conserved(B)\( K_A + U + E_{int} \) is conserved(C)\( K_A + K_B + E_{int} \) is conserved(D)\( K_A + K_B + U \) is conserved(E)\( K_A + K_B + U + E_{int} \) is conserved

7. A uniform stick of length l is pivoted at one end, and released from rest in a horizontal position. How long will it take for it to swing down to the vertical position?(A)\( 1.311\sqrt{\frac{l}{3g}} \)(B)\( 2.622\sqrt{\frac{l}{3g}} \)(C)\( 1.311\sqrt{\frac{l}{2g}} \)(D)\( 2.622\sqrt{\frac{l}{2g}} \)(E)\( 2.622\sqrt{\frac{l}{g}} \)

8. The displacement of a string is given by \( y(x,t) = y_m \sin(kx + \omega t) \). The speed of the wave is(A)\( \frac{2\pi k}{\omega} \)(B)\( \frac{\omega}{k} \)(C)\( \omega k \)(D)\( \frac{2\pi}{k} \)(E)\( \frac{k}{2\pi} \)

9. A diffraction-limited telescope, designed for use in space, has an aperture of 1 m and a focal length of 20 m. At the focus in a TV detector whose "resolution element" size is \( 10^{-5} \) m. At what wavelength is the telescope optimal in the sense that \( 10^{-5} \) m corresponds to the diffraction limit?(A)4,000 Å(B)2,000 Å(C)1,000 Å(D)3,000 Å(E)5,000 Å

10. An ideal monatomic gas at room temperature is adiabatically decompressed so that the final volume is 8 times the original. What's the final temperature?(A)55 K(B)75 K(C)150 K(D)85 K(E)100 K

11. How long would it take for a 1000 W heater to melt 1.00 kg of ice at −20.0°C, assuming all the energy from the heater is absorbed by the ice?(A)4.18 s(B)41.8 s(C)5.55 min(D)6.25 min(E)38.4 min

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