Do not fill out the information below until instructed to do so! Name: Signature: Student ID: E-mail: Section Number: Formulae are provided on the last page. You may NOT use any other formula sheet. You may use this exam or come up front for scratch paper. Although the questions are multiple-choice all work must be shown to get credit for the answer marked. If the answer marked does not obviously follow from the shown work, even if the answer is correct, you will get zero credit for the answer. The points per question are indicated. Clearly erase any unwanted marks. No credit will be given if we can t figure out which answer you are choosing. Be careful in marking your answer. Because it is multiple-choice format there will be no partial credit. If you mistakenly mark the wrong answer you will get zero credit for the answer. Double check your work and answers before turning them in. Each question is worth 5 points unless otherwise marked. Put your initials here after reading the above instructions (doing so means that you understand and agree with the above instructions):
Two identical masses are released from rest in a smooth hemispherical bowl of radius R, from the positions shown in the figure. You can ignore friction between the masses and the surface of the bowl. 1. What is the velocity of the first block before it strikes the stationary block at the bottom? a) b) c) d) gr gr /2 gr 2gR 2. If they stick together when they collide, how high above the bottom of the bowl will the masses go after colliding? a) b) c) d) 2R R R /4 2R
Block A in the figure has mass 1.00 Kg, and block B has mass 3.00 Kg. The blocks are forced together, compressing a spring S between them; then the system is released from rest on a level, frictionless surface. The spring, which has negligible mass, is not fastened to either block and drops to the surface after it has expanded. Block B acquires a speed of 1.40 m/s. 3. What is the final speed of block A? a) 4.20 m/s b) 1.40 m/s c) 0.47 m/s d) 2.30 m/s e) 0 m/s 4. How much potential energy was stored in the compressed spring? a) 1.98 J b) 3.00 J c) 0 J d) 11.76 J e) 5.4 J
An electric turntable 0.700 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.280 rev/s. The angular acceleration is 0.899 rev/s 2. 5. Compute the angular velocity after a time of 0.197 s. a) 0.341 rev/s b) 1.45 rev/s c) 0.752 rev/s d) 0.457 rev/s e) 0.122 rev/s 6. Through how many revolutions has the blade turned in this time interval? a) 0.073 rev b) 0.280 rev c) 1.02 rev d) 0.0 rev e) 0.146 rev 7. What is the tangential speed of a point on the tip of the blade at time t = 0.197 s? a) 1.50 m/s b) 2.00 m/s c) 3.14 m/s d) 1.00 m/s e) 0.25 m/s 8. What is the magnitude of the resultant acceleration of a point on the tip of the blade at time t= 0.197 s? a) 1.98 m/s 2 b) 2.89 m/s 2 c) 3.50 m/s 2 d) 1.75 m/s 2 e) 0.0 N
It is well known that for a hollow, cylindrical shell rolling without slipping on a horizontal surface, half of the total kinetic energy is translational and half is rotational. What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface? 9. A uniform solid cylinder. a) 1/2 b) 1/3 c) 2 d) 4 e) 1/4 10. A uniform sphere. a) 2/7 b) 2/5 c) 7/2 d) 5/13 e) 1/5 11. A thin-walled hollow sphere. a) 2/7 b) 2/5 c) 7/2 d) 5/13 e) 1/5 12. A hollow, cylinder with outer radius R and inner radius R/2. a) 3/4 b) 2/3 c) 3/2 d) 5/13 e) 1/2
Two metal disks, one with radius R 1 = 2.56 cm and mass M 1 = 0.850 Kg and the other with radius R 2 = 4.88 cm and mass M 2 = 1.55 Kg, are welded together and mounted on a frictionless axis through their common center.. 13. What is the total moment of inertia of the two disks? a) 2.12 x10-3 m 2 kg b) 21.2 m 2 kg c) 4.24 x10-3 m 2 kg d) 4.24 m 2 kg e) 3.14 x10-3 m 2 kg 14. A light string is wrapped around the edge of the smaller disk, and a 1.50-kg block, suspended from the free end of the string. If the block is released from rest at a distance of 2.10 m above the floor, what is its speed just before it strikes the floor? a) 2.43 m/s b) 3.20 m/s c) 5.00 m/s d) 2.50 m/s e) 3.61 m/s 15. Repeat the calculation of part B, this time with the string wrapped around the edge of the larger disk. a) 2.43 m/s b) 3.20 m/s c) 5.00 m/s d) 2.50 m/s e) 3.61 m/s
For the yo-yo shown in the figure. 16. Find the tension in the rope a) Mg/2 b) Mg/3 c) Mg d) 2Mg e) 1.5 Mg 17. Find its acceleration a) g/3 b) 2g c) g/2 d) g e) 2g/3
18. Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly 10 14 times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was 7.0 10 5 Km (comparable to our sun); its final radius is 18 km. If the original star rotated once in 35 days, find the angular speed of the neutron star. a) 1.14 x 10 3 rad/s b) 2.14 x 10 3 rad/s c) 3.14 x 10 3 rad/s d) 4.14 x 10 3 rad/s e) 5.14 x 10 3 rad/s
The horizontal beam in the figure weighs 150 N, and its center of gravity is at its center. 19. Find the tension in the cable. a) 350 N b) 75 N c) 625 N d) 500 N e) 250 N 20. Find the horizontal component of the force exerted on the beam at the wall. a) 350 N b) 75 N c) 625 N d) 500 N e) 250 N 21. Find the vertical component of the force exerted on the beam at the wall. a) 350 N b) 75 N c) 625 N d) 500 N e) 250 N
F = ma F B on A = F f static µ s F N, W by F = r 2 r 1 F A on B F dr Formula sheet F f kinetic =µ k F N F spring = F constant Δ r = F constant Δrcosθ KE = 1 2 m v2 PE g = mgy PE spring = 1 2 kx 2 = kx P = dw dt = F v W total = KE 2 KE 1 W non -conservative forces = E 2 E 1 E = KE + PE g + PE spring Vectors A = A xˆ i + A ˆ y j + A ˆ z k A B = ABcosθ A B = A x B y + A y B y + A z B z The following always apply v x = dx dt a x = d v x dt v x average = x(t ) x(t ) 2 1 t 2 t 1 a x average = v(t ) v(t ) 2 1 t 2 t 1 The following apply for constant acceleration x x 0 = v x 0 t + 1 2 a x t 2 Other equations a c = v2 R = 4π 2 R T 2 Some constants g = 9.80 m s 2 A = A = A 2 x + A 2 2 y + A z v x = v x 0 + a x t v = d r dt a = d v dt v P / A = v P / B + 1 km hr = 0.278 m s Some mathematical formulas: if x(t) = at n dx dt = nat n 1 v B / A A B = ABsinθ A B v 2 2 x = v x 0 + 2a x (x x 0 ) at 2 + bt + x = 0 t = b + b2 4ac 2a
Moment of inertia of different objects Formula sheet (continued) Hollow cylinder Solid cylinder I = 1 2 M(R 2 1 + R 2 2 ) I = 1 2 MR2 Thin-walled hollow cylinder/ring I = MR 2 Solid Sphere I = 2 5 MR2 Thin walled sphere KE rot = 1 2 Iω 2 τ = r τ = I F α L = r mv = Iω The following apply for constant angular acceleration θ θ 0 = ω 0 t + 1 ω = ω 0 + αt 2 αt 2 ω 2 = ω 2 0 + 2α(θ θ 0 ) Other equations a tan = αr v tan = ωr I = 2 3 MR2 KE = 1 2 mv 2 1 cm 2 I cmω 2