2010 Fall UCONN 1201 PROBLEMS

  1. A man stands on a platform that is equipped with rollers. The man holds a rifle aimed horizontally, with the rifle butt braced against the man’s shoulder. The man and the platform are both motionless. As part of a ballistics test, the man then fires the rifle so that the bullet travels horizontally in one direction and the man and platform move in the opposite direction to the bullet. Immediately after the bullet leaves the gun, the man and rolling platform have kinetic energy KE1 and the bullet has kinetic energy KE2. The bullet has mass m and the man plus platform have mass M. Find the ratio R, as given by R = (KE2 / KE1) in terms of M and m.
  1. A box is placed on the outer rim of a horizontal disk that has an axle through its center. The disk is set to be spinning at constant angular speed with a period of 24 seconds. The radius of the disk is 4.6 m. How large must the coefficient of static friction be between the box and the disk for the box to stay on the rotating disk without slipping?
  1. An oil drilling machine rests on the ocean floor about 5000 ft below the ocean surface. It drills

through an additional 13,000 feet of soil when there is a failure and oil escapes through some

kind of rupture. Assume the drill hole has a circular base consisting of the drill extending

through a tall column of muddy paste, and that the diameter of this circular base is 18 cm.(a)

Compute the approximate hydrostatic pressure at the base of the drill hole just before the

failure. Assume the density of water is 1000 kg/m3 and the density of the soil is 2.5 times that

of water. (b) What would be the force on the circular base of the drill hole at this pressure?

  1. Lake Champlain has a surface area of 1,126 km2. It freezes in the winter with an average ice thickness of 0.5 m. (a) What total energy does it take to melt the ice once spring comes? (b) If this energy is provided by sunlight shining on the lake at 200 W/m2, and this energy is only half absorbed, how long will it take the ice on the lake to melt?
  1. A geosynchronous satellite stays above the same point on the equator as the earth revolves. Given the mass of the earth M, and gravitational constant G, find the distance D of the satellite from the earth’s center.
  1. An oil spill on the ocean is sometimes modeled as having a velocity that has two components: a wind induced velocity that is 3% of the true wind velocity and an independent tidal velocity Vtidal. The tidal velocity is 3.0 km/hr due north and V wind = 40 km/hr 20 degrees north of east. In six hours, how far is the oil predicted to move in the northerly direction?

  1. The above diagram illustrates the simple mass on a spring accelerometer. Note that were the acceleration to the right, instead of the left, the spring would compress rather than stretch. Let there be a vehicle with a flat top on which such an accelerator is fastened. The vehicle moves horizontally at all times and the accelerometer is also mounted horizontally as shown above. The mass on the spring is m, the spring constant is K, and the mass of the vehicle is M, where M > m. The length of the unstretched spring is D, and the vehicle is moving at constant velocity to the west. Gradually the brakes are applied, and then held at constant braking force. The length of the spring is now observed to be L = D – b where L, D, and b are all positive numbers. What is the braking force on the vehicle in terms of given quantities?
  1. For the statics problem illustrated above, a strut is hinged at point q, a guy wire is attached to the strut at point p, and the wire is attached also to the horizontal surface along the X axis. A mass m hangs from point p, the mass of the strut is M and the strut is uniform, and the angles are shown. The mass m = 225 kg and mass M = 45 kg. What is the tension T in the wire stretched between o and p?
  1. Give a statement of the work – energy theorem in words.
  1. Give a statement of the law of conservation of momentum in words.
  1. Give a statement of Newton’s universal law of gravitation in words
  1. Give one of the statements of the second law of thermodynamics in words.

13.

Given masses M2 and M1 being dragged across the frictionless table by a force F from a string. There is a massless rope connecting the mass. Derive an expression for the tension T between the masses, in terms of F, M2, and M1.

Answers to U. Conn. Problems

1.Let V be the bullet velocity and U be the velocity of man plus platform.

(1)mV = MU,

(2)U = mV/M

(3)MU2 / 2 = KE1 = (M/2)(m2V2 /M2)

(4) (1/2)=mV2/2 = KE2

From 3 and 4 above KE2/KE1 = R = M/m

2. The centripetal force on the box is given by the expression:

(1)FC = MV2/R where M is assumed to be the mass of the box. Also one has:

(2)FC ≤ μ Mg where μ is the coefficient of static friction. Equate expressions (1) and (2); The M factors cancel and one is left with:

(3)μ ≥ V2 / R g =[ (2πR)2/T2]/R g = 4π2R/T2 g..Substituting, one obtains μ ≥ 0.032

3.The pressure due to the column of ocean water is given by

(1)P W = ρ W g h W = 1000 kg/m3 x 9.8 m/sec2 x h W ; take one yard to be a meter and three feet. Then P W = 167 x 105 Pa ~ 167 atm. Neglect the atmospheric pressure at the surface of the ocean as completely negligible for purposes of this situation.

(2)P soil = ρ soil g h soil ~ 1083 atm

(3)P net = P soil + P w ~1250 atm; the industry at least in the U.S. commonly uses 1 atm as 14 pounds/sq inch and this comes out then to about 17500 pounds / sq inch. At 2.54 cm/inch, the area A of the drilling cylinder is 39 inch2. Then F = PA = 17500 pounds/sq inch x 39 inch2 ~ 7 x 105 pounds

4.The density of ice is 920 kg/m3 and the heat of fusion of ice is 334 kj/kg. Then M = density x volume = 5.2 x 1011 kg. Then the energy required is E = 5.2 x 1011 kg x 334 kj/kg = 170 x 10 15 J

Total power = power/area x area = 200 w/m2 x 1,126 x 106 m2 = 2.25 x 10 11 watts

P = E/t, so t = E /P = 170 x 10 15 J / 2.2f x 1011 W = 76 x 10 4 sec , BUT energy from the sun is only half absorbed so it really takes twice this long.

Therefore t =152 x 104 seconds, or 422 hours Answer is 18 days

5. Let m be the mass of the satellite. Then one has:

mV2/D = GmM/D2

V2 = GM/D

But V = 2πD/T

4π2D2/T2 = GM/D

D3 = (GM/4π2) T2

T must equal one day for the satellite to exactly keep up with the earth’s rotation.

T = 24 hours x 3600 sec/hour = 86,400 sec

D where T = 86,400 sec

6. V tidal = 3.0 km/hr due north and V wind = 40 km/hr 20 degrees north of east.

In six hours, how far is the oil predicted to move in the northerly direction?

Answer: V wind induced = 0.03 x V w i = 1.2 km/hr 20 degrees north of east. Draw these vectors approximately to scale:

The total northerly component of velocity = 3 km/hr + (1.2 sin 200) km / hr = 3.4 km/hr.

D north = 3.4 km/hr x 6 hr = 20.4 km

7. Let f be the force compressing the spring. F = Kb. The acceleration of m is then a = - Kb/m if + direction is west. But this is also the acceleration of the vehicle, since the accelerometer is bolted to the vehicle’s top. Therefore, F = Ma = - (M K b)/m

8.Since the sum of the angles in a triangle is 180 degrees, angle poq = 135 degrees. Take sum of torques about q = zero.

Assume the length of the strut is L. This will eventually cancel out.

C.W. torques: Mg L/2 cos 450 + mg L cos 450

C.C.W. torques: the moment arm for T is M = L sin 150

Therefore, C.C. W torques = TM = TLsin 150 Equating C.W. and C.C. W. one obtains

T = ((Mg/2) cos 450 + mg cos 450) / sin 150 Substituting, T = 6.64 x 103 N

9.

10.(1) T = M2 A; (2)F – T = M1 A ; add these to obtain (3)F = (M2 + M1 ) A

(4) A = F / (M2 + M1) ; substitute 4 into 1

T = M2 ( F/(M2 + M1) )