Exercises for PHYS 2325

Dr. Cox

rev. Feb. 2007

These Exercises were assigned in the summer a few years ago. Some of them are problems that have also been, or could have been, at other times, quiz problems. Others are more time-consuming, or might call for referring to tables, and so would not be considered for quiz use.

Some typo or formatting problems may appear, possibly varying with browser.

The following rules apply if Exercises are to be turned in for a grade:
1) Use loose pages, one side only, unfolded. Number all pages in the upper right corner according to the pattern that 3/6 means third page of six.
2) You are to work on exercises in groups. Turn in one answer per group, showing all names on each page, and all will get the same credit.
3) You may use any resources, including professors, to help you understand the questions, provided it is understood by any people you consult that these count toward a grade and that it is your understanding and not the result that should count.
4) Show enough steps that your work is understandable; more than one page for a problem is in most cases too much detail unless your writing is large. Any incorrect details in any step will be penalized; so will complete absence of detail. Partial credit will be given to a limited extent when earned by legible relevant work, but may be lost if accompanying errors are more significant. Not following these rules will also be penalized even if the physics is correct.
5) Units are part of any correct physical quantity.

The following information, at least, is made available on the course final exam; appropriate portions are presented for quizzes.
Use the following values if needed:
1 foot = 0.30 meter ;     1 hp = 746 watt (W) ;     1 cal = 4.2 joule
Earth's gravitational field: 9.80 m/s² ;       Newton's gravitational constant: 6.67x10^-11 N m²/kg²
Some moments of inertia (about centers):  disk: (1/2)MR² ;   sphere: (2/5)MR²
Standard atmosphere: 1.013x10⁵ N/m² ;     Absolute zero: -273 ^oC
Density of water: 1000 kg/m³ ;     Density of air: 1.29 kg/m³ (standard temperature, pressure)
Some properties of steel:   specific gravity: 7.9 ;       Young's modulus: 2.0x10¹¹ N/m²
      shear modulus: 0.84x10¹¹ N/m² ;     bulk modulus: 1.6x10¹¹ N/m²
Coefficients of linear expansion:   aluminum: 25x10^-6/C^o ;     steel: 12x10^-6/C^o
Specific heat capacity:   water: 1 cal/(g C^o) ;     ice: 0.5 cal/(g C^o)
      steam: 0.5 cal/(g C^o) ;     aluminum: 0.2 cal/(g C^o) ;     copper: 0.1 cal/(g C^o)
Thermal conductivity:  of water: 0.14 cal/(m s C^o) ;     of copper: 92 cal/(m s C^o)
Latent heats, water:  fusion: 80 cal/g ;     vaporization: 540 cal/g

Reminder: a problem you don't read correctly is a problem you will not do correctly.

Exercise 1, on motion.
It is 2304 and space cadets are training at the world Ceres, where gravity is negligible. They use a coordinate system with origin at Ceres' surface, y axis up. Today, Carmen is correctly executing a series of maneuvers. At t = 0, three minutes after maneuver 5 ends, she is crossing the y axis, 20 km up from the surface, moving at - 400 k m/s, and she begins the 6th maneuver, which consists of 40 s of constant acceleration, to be followed (after 90 s of no acceleration) by maneuver 7. At t = 30 s her velocity is 500 m/s in a direction 36.9^o from -k toward +j.
1. Find her velocity at t = 2 minutes.
2. Find her position components at t = 20 s.
3. Find her position at t = 2 minutes.

Exercise 2, on forces and Newton's Laws
1. A system of ropes is holding the Joe's Diner sign (200 N) in place. One runs straight up from the sign to a knot; from the knot, one runs due north to a tree while the other, running to his roof edge, makes an angle of 30^o with the vertical. List, in a neat table, all the forces on (and only those on) the knot. (This requires giving suitable identification, not value, elements.)
2. A chest (mass 50 kg) is sitting on a level floor. Peggy (mass 40 kg) will apply a force of 60 N westward, trying to move it, while Nan (mass 60 kg) is set up to apply a southward force of 80 N. The two girls start to push simultaneously and continue for several seconds; identify, in a neat table, all the forces on (and only those on) the chest after 1.0 s.
3. A model train consists of five cars and the engine; the cars are numbered in order beginning with #1 next to the engine. Each car has a mass of 1.5 kg, and the model has been accelerating southward on level track, led by the engine unit, at 2.3 m/s² since it started from rest 2.0 s ago. Identify, in a neat table, all the forces on (and only those on) car 2.
4. In problem 1, find the tension in the northward rope; assume masses are negligible for all the ropes. Indicate your reasoning.
5. In problem 2, find the displacement of the chest after 1.0 s; assume the floor is slick. Indicate your reasoning.
6. In problem 3, find the force that will be exerted by car #3 on car #2 if friction on the cars by the track is negligible. Indicate your reasoning.

Exercise 3, more on forces and Newton's Laws
1. A model racecar, mass 12.0 kg, is operating at 1.5 m/s on a uniform circular track of radius 0.5 m which is banked at 36.9^o from the horizontal. The curve is always to the left, and at this moment the car is moving east. What is the vector friction force on the car?
2. A chest (mass 50 kg) is sitting on a level floor. Peggy (mass 40 kg) will apply a force of 60 N westward, trying to move it, while Nan (mass 60 kg) is set up to apply a southward force of 80 N. The two girls start to push simultaneously and continue for several seconds. If friction with the floor is described by a static coefficient of 0.25 and a kinetic coefficient of 0.15, find the friction force and also the displacement of the chest after 1.0 s.
3. A model train consists of five cars and the engine; the cars are numbered in order beginning with #1 next to the engine. Each car has a mass of 1.5 kg, and the model has been accelerating southward on level track, led by the engine unit, at 2.3 m/s² since it started from rest 2.0 s ago. If friction exerts 3.0 N on each car, find the force exerted by car #3 on car #2.

Exercise 4, on work and energy
A horizontal spring (force constant 240 000 N/m, unstretched length 20.0 cm), is compressed to a length of 13.0 cm, and a 12-kg steel block is placed at rest next to it. When the spring is released, the block is pushed away and comes to an upward-sloping surface; the block comes to rest on that surface at a height of 4 m above its starting point. The block slides freely until just after the surface begins to rise.
1. Neatly and properly identify all the forces which act on the block between release and coming to rest.
2-4. Determine the amount of work done, during the motion from release to coming to rest, by each force acting on the block. (This part carries multiple credit because of there being multiple forces [but not necessarily 3] and therefore multiple calculations, requiring multiple approaches.)
5. If the slope is at an angle of 36.9^o, determine the average coefficient of friction between block and slope. Will the block slide back downhill, if friction is uniform?

Exercise 5, on impulse and momentum
1-2. Dorothy (30 kg) and the Wizard (50 kg) were skating north together across a smooth patch of Winterland ice at 2.0 m/s when they were surprised by the Cowardly Lion (40 kg) jumping from behind a bush. The lion, moving west at 5.0 m/s, panicked at finding himself sliding frictionlessly, and pushed so hard on his friends that he was left sliding east at 1 m/s while Dorothy was sliding south at 3.0 m/s.
      1. What was the Wizard's velocity just after this event?
      2. What was the change in the three-person system kinetic energy?
3. A game set consists of a board and four playing pieces. The board (0.60 kg) is a 40 cm square, marked as the coordinate system with origin at its lower left corner and axes along its edges; the pieces (100 g each) are now on it at (30 i + 10 j) cm, at (20 i + 35 j) cm, at (35 i + 15 j) cm, and at (5 i + 30 j) cm. Where is the center of mass of this five-piece system?

Exercise 6, on rotation and related chapters
1. A carousel (a powered merry-go-round), 5.0 m in diameter, started braking 10 s ago; in a total of 15 s of uniform change it will have stopped from its "full speed" motion of 6 revolutions per minute. You are riding the carousel, 2.0 m east of its axis, going north, and a fly (20 g) landed on the top of your head 1 second ago and is riding along with you. Find the total contact force by your hair (assuming you're not bald) on the fly now.
2. A force is given by the function F = (2 (x+y) i - 3 (y+z) j) N/m. (That is, the value of this particular force on any particle depends on where that particle is, according to the formula, which is in terms of the particle's position components, or coordinates.) A 3.0-kg mass is at (-4 m i + 2 m k), and the center of rotation is at 2 m i. Find the torque about that center provided by this force on that particle.
3-4. (A double-value problem) Alice's Restaurant has to replace its sign. The new sign is a uniform rectangle (3 m wide, 2 m high, 500 N). The area below is planted with roses, so the sign is presently being held in place (in equilibrium, not necessarily in stable equilibrium) by two ropes. One rope is tied to the upper north corner of the sign, while the other rope runs due south from the bottom center point of the sign. Find the vector force exerted by each rope on the sign.

Exercise 7, on several chapters
1. (Elasticity) Little Peter has a rectangular portion of Jello on his plate. It is 2.0 cm high, 5.0 cm long, and 4.0 cm wide. He is applying a force of 45 N to its top surface, in the direction parallel to the longest side, as a result of which the Jello leans 0.60 mm. Which elastic modulus is tested here, and what is its average value?
2. (Gravity) Captain Hook (mass 100 kg) is investigating a new star system in his scout rocket (mass 8000 kg). The star (mass 6.0x10²⁹ kg) is at the origin, while his rocket is at (-8.0x10⁶ km i + 6.0x10⁶ km k). What is the vector gravitational force on his rocket?
3. (Fluid at rest) A 14-cm-long sealed cylindrical tube is floating upright in a lab water tank which incorporates an artificial oil slick. The tube's end area is 2.0 cm²; it has 4.0 cm of its length below water level and another 5.0 cm in the oil (density 800 kg/m³). What is the weight of the tube?
4. (Fluid in motion) Water is flowing into a house through a 12.0-cm² pipe in the basement. Upstairs, 5.00 m higher, it is flowing smoothly through a 4.00-cm² pipe at 6.00 m/s under a pressure of 4.00x10⁴ Pa. If no other flows are involved, what is the pressure in the basement pipe?
5-6. (Thermal expansion, Heat capacity) A hollow block of hot copper (temperature 170 ^oC, 8000. cm³) is put in a closed plastic (insulating) container partly filled by 500 g of cool water (temperature 16 ^oC), along with a 10-g ice cube (temperature -10 ^oC). (Any steam that forms is not allowed to escape.) Soon the block and the water are all at a temperature of 20 ^oC.
      5. What is the mass of the copper?
      6. What is the final volume of the copper block?

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