Physics 125 Test 2 April 2, 2020 Name Part 1: Short Question

Physics 125 Test 2 Apr 2 2020 Name Part 1 Below Are 6 Short

Below are 6 short workout questions. Answer any 5 of the 6 questions. YOU MUST SHOW ALL YOUR WORK TO RECEIVE ANY CREDIT. You may do the remaining question as a bonus.

Indicate which questions are to be graded and which is the bonus.

Part 1: Short Workout Questions

1. a) A bomb is moving to the right at 0.3 c. Ship A is chasing the bomb at 0.5 c to the right while ship B is chasing the bomb at 0.5 c to the left. Which ship, A or B, will measure the largest dilated time for the bomb to explode? Explain your reasoning.

2. b) Three equal mass objects are each acted upon by different forces acting in the x-direction. The x-velocity as a function of time for each mass is shown below. Rank the forces in the x- direction acting on each of the masses from smallest to largest. Note any ties.

3. c) Mass m is at rest on an inclined plane. The coefficient of static friction between the mass and the inclined plane is μs. Student A claims that the static frictional force acting on m is equal to μsF. Student B claims that the static frictional force acting on m is equal to the component of the weight parallel to the incline. Student C claims that the static frictional force acting on m must exceed the x-component of the weight. Which student is correct? Explain your reasoning.

4. d) Mass m is swung in a vertical circle of radius R. Rank the tensions in the string at points A, B, and C from least to greatest assuming the speed is constant. Do not neglect gravity.

5. e) Find an expression for the acceleration of the masses. Your answer should only contain m*, m+, μs, and various constants.

6. f) Masses m and m+ are connected by a string passing over a pulley. Mass m is at rest on an inclined plane with coefficient of static friction μs. What is the maximum value that m+ can have before m* starts to move? Provide an expression.

7. g) A car of mass m is traveling over hills and valleys modeled as hemispheres of radius R. (a) Determine the speed at which the car just loses contact at the top of the hill. (b) Find the normal force at the bottom point B if the initial speed is maintained.

8. h) You measure the length of spaceship A as 200 m. You observe spaceship B approaching at 0.99c. What is the length of spaceship B as measured by someone on spaceship B? Additionally, what is the length of spaceship A as measured by someone on spaceship B?

9. i) A ball is thrown upward with initial velocity v0. Assuming air resistance follows F = -k v, prove that the velocity as a function of time is v(t) = (v0 + (k/m)) e^{-(k/m)t} - (k/m) v0, starting from the free body diagram and considering acceleration.

Part 2: Longer Workout Questions

1. 1. An astronomer on Earth discovers planet X 100 light-years away. A ship travels at 0.85 c towards the planet. The inhabitants of planet X shoot a missile at the ship at 0.9 c.

   • a) Find the distance between Earth and planet X according to someone on the Earth ship and someone on the missile.
   • b) Determine the velocity of the missile according to someone on the ship from Earth.
   • c) If the missile self-destructs in 2 hours according to the Earth ship, how long until self-destruction according to someone on the missile?

2. 2. Three masses connected as shown: masses m1, m2, m3, with inclined plane at angle q and coefficient of kinetic friction μk. Pulleys are massless and frictionless. Do not solve the equations yet; just set them up for each mass using Newton's second law.

3. 3. Two 1 kg balls connected by strings swung in horizontal circles. Find the tension in each string and velocities, considering gravity, for given radii and angles.

Instructions Summary

Show all work to receive credit. Provide clear, detailed solutions including diagrams where necessary. Use correct physics principles and notation. Write in full sentences with explanations. Where calculations are involved, show each step clearly and specify the relevant equations used.

Paper For Above instruction

The questions in this test cover diverse areas of physics, emphasizing both conceptual understanding and quantitative problem-solving. They span special relativity, classical mechanics, and rotational dynamics, requiring students to understand relativistic effects, Newton's laws under various forces, and dynamics of rotational motion in different contexts.

Starting with the relativistic questions, problem 1 involves understanding time dilation from different inertial frames. The key is recognizing that the observer moving closer to the event (the bomb explosion) will measure a different dilated time. Electronically, this is handled via the Lorentz factor γ, which depends on the relative velocity between observer and event. The ship chasing the bomb at 0.5 c will observe a different dilated time compared to the ship moving at 0.3 c, with the Doppler effect influencing the measurement, particularly if the event is moving toward or away from the observer. Theoretical formulas include:

Δt' = γ Δt

where γ = 1 / √(1 - v²/c²). The larger the relative velocity, the greater the Lorentz factor and hence the greater the dilation.

In the classical mechanics section, forces and motion equations come into play. For example, analyzing the forces causing acceleration of masses attached via pulleys or moving on inclined planes requires applying Newton's second law, considering frictional forces, tension, component forces along slopes, and constraints imposed by pulleys.

The problem involving the inclined plane and static friction involves understanding that static friction adjusts up to a maximum value of μsN, where N is the normal force, and the actual static friction is equal to the component of weight parallel to the incline until impending motion occurs. The maximum static friction force before motion begins is μsN.

The rotational dynamics problem asks for the tension at different points on a vertical circle. The tension varies with position because of the combination of centripetal acceleration and gravity. Typically, at the top of the circle, tension minimizes, and at the bottom, tension maximizes; ranking these using the tension formula T = m(v²/R) - mg (top), T = m(v²/R) + mg (bottom).

In the gravitation and planetary motion sections, the focus is on relativistic length contraction and effects in space travel, requiring knowledge of Lorentz contraction: L = L₀/γ.

The air resistance problem involves differential equations reflecting forces proportional to velocity, leading to exponential decay solutions for velocity and position functions. This requires solving first-order linear ODEs with appropriate initial conditions.

Overall, the questions require combining classical mechanics, circular motion, rotational dynamics, and special relativity principles, highlighting the importance of understanding the physical laws governing diverse systems.

References

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