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1. Question: (7 points, suggested time 13 minutes)

A wooden wheel of mass M, consisting of a rim with spokes, rolls down a ramp that makes an angle θ with the horizontal, as shown above. The ramp exerts a force of static friction on the wheel so that the wheel rolls without slipping.

(a)

i. On the diagram below, draw and label the forces (not components) that act on the wheel as it rolls down the ramp, which is indicated by the dashed line. To clearly indicate at which point on the wheel each force is exerted, draw each force as a distinct arrow starting on, and pointing away from, the point at which the force is exerted. The lengths of the arrows need not indicate the relative magnitudes of the forces.

ii. As the wheel rolls down the ramp, which force causes a change in the angular velocity of the wheel with respect to its center of mass?

Briefly explain your reasoning.

(b) For this ramp angle, the force of friction exerted on the wheel is less than the maximum possible static friction force. Instead, the magnitude of the force of static friction exerted on the wheel is 40 percent of the magnitude of the force or force component directed opposite to the force of friction. Derive an expression for the linear acceleration of the wheel’s center of mass in terms of M, θ, and physical constants, as appropriate.

(c) In a second experiment on the same ramp, a block of ice, also with mass M, is released from rest at the same instant the wheel is released from rest, and from the same height. The block slides down the ramp with negligible friction.

i. Which object, if either, reaches the bottom of the ramp with the greatest speed?

________Wheel ________Block _________Neither; both reach the bottom with the same speed.
Briefly explain your answer, reasoning in terms of forces.

ii. Briefly explain your answer again, now reasoning in terms of energy.

Answer/Explanation

Ans:

(a) (i)

(ii)

The fork of static friction.

Briefly explain your reasoning.

The static friction force is being applied at a point at a radius with respect to the enter of mass, causing a torque and subsequent change in angular velocity.

(b) \(\sum F = \frac{ma}{m} = \frac{Mgsin\Theta – 0.4 Mgsin\Theta }{M}\)

      a =  0.6 gsh

(c)

i. _X_ Block

The block only has a single force acting upon it gravity. The wheel has two gravity and friction, which is opposing the downward motion. Since friction is slowing the wheel down, the block of ice will roach the bottom of this range with the greatest speed.

ii.

All of the block’s gravitational potential energy is being converted directly into linear kinetic energy. The wheel’s GPE is being converted into both linear kinetic energy and rotational kinetic energy. Since both objects have the some starting GPE, the block will reach the bottom with the greatest velocity because all of its GPE is being converted to linear KE.

2. Question: (12 points, suggested time 25 minutes)

A new kind of toy ball is advertised to “bounce perfectly elastically” off hard surfaces. A student suspects, however, that no collision can be perfectly elastic. The student hypothesizes that the collisions are very close to being perfectly elastic for low-speed collisions but that they deviate more and more from being perfectly elastic as the collision speed increases.

(a) Design an experiment to test the student’s hypothesis about collisions of the ball with a hard surface. The student has equipment that would usually be found in a school physics laboratory.

i. What quantities would be measured?
ii. What equipment would be used for the measurements, and how would that equipment be used?
iii. Describe the procedure to be used to test the student’s hypothesis. Give enough detail so that another student could replicate the experiment.

(b) Describe how you would represent the data in a graph or table. Explain how that representation would be used to determine whether the data are consistent with the student’s hypothesis.

(c) A student carries out the experiment and analysis described in parts (a) and (b). The student immediately concludes that something went wrong in the experiment because the graph or table shows behavior that is elastic for low-speed collisions but appears to violate a basic physics principle for high-speed collisions.

i. Give an example of a graph or table that indicates nearly elastic behavior for low-speed collisions but appears to violate a basic physics principle for high-speed collisions.
ii. State one physics principle that appears to be violated in the graph or table given in part (c)i. Several physics principles might appear to be violated, but you only need to identify one.

Briefly explain what aspect of the graph or table indicates that the physics principle is violated, and why.

Answer/Explanation

Ans:

(a)

i.

height of boll before collision

height of boll after collision

ii.

A tape measure would be used to measure the height of the boll at its peak before and after the collision.

iii.

  1. Drop the ball at a height of 1 foot determined by tape measure and record height with the same tape measure.
  2. Repeat 3 times.
  3. Repeat 1 – 2 with heights 2ft, 3ft, 5ft, 10ft, and 15ft.

(b) The X axis would be the height dropped from and the Y axis would be the new height. If the collision is elastic the data should be linear at a lil ratio of the before and after heights. However, if the students hypothesis is correct the data should deviate downward from this trend line at higher heights because the greater the height the mole velocity produced upon collision.

(c)

i. 

ii. conversation of energy is being violated because the ball is getting higher after the collision than from when it was dropped. Because mass and acceleration due to gravity are constant that means the ball gained more energy after the collision and did not conserve it.

3. Question: (12 points, suggested time 25 minutes)

A long track, inclined at an angle θ  to the horizontal, has small speed bumps on it. The bumps are evenly spaced a distance d apart, as shown in the figure above. The track is actually much longer than shown, with over 100 bumps. A cart of mass M is released from rest at the top of the track. A student notices that after reaching the 40th bump the cart’s average speed between successive bumps no longer increases, reaching a maximum value vavg . This means the time interval taken to move from one bump to the next bump becomes constant.

(a) Consider the cart’s motion between bump 41 and bump 44.
i. In the figure below, sketch a graph of the cart’s velocity v as a function of time from the moment it reaches bump 41 until the moment it reaches bump 44.
ii. Over the same time interval, draw a dashed horizontal line at v = vavg. Label this line “ vavg ”.

(b) Suppose the distance between the bumps is increased but everything else stays the same.
Is the maximum speed of the cart now greater than, less than, or the same as it was with the bumps closer together?
_______Greater than ________ Less than _________The same as
Briefly explain your reasoning.

(c) With the bumps returned to the original spacing, the track is tilted to a greater ramp angle θ . Is the maximum speed of the cart greater than, less than, or the same as it was when the ramp angle was smaller?
_________ Greater than __________ Less than ___________The same as
Briefly explain your reasoning.

(d) Before deriving an equation for a quantity such as vavg, it can be useful to come up with an equation that is intuitively expected to be true. That way, the derivation can be checked later to see if it makes sense physically. A student comes up with the following equation for the cart’s maximum average speed:   vavg = \(C\frac{Mgsin\theta }{d}\), where C is a positive constant.

i. To test the equation, the student rolls a cart down the long track with speed bumps many times in front of a motion detector. The student varies the mass M of the cart with each trial but keeps everything else the same. The graph shown below is the student’s plot of the data for vavg as a function of M.

Are these data consistent with the student’s equation?
_________Yes ________No
Briefly explain your reasoning.

ii. Another student suggests that whether or not the data above are consistent with the equation, the equation could be incorrect for other reasons. Does the equation make physical sense?
_________Yes ____________ No
Briefly explain your reasoning.

Answer/Explanation

Ans:

(a)

(b)

√  Greater than

Since cart will travel a greater distance there will be more time for the cart to accelerate ergo  it will be able to attain a higher speed.

(c)

√  Greater than

With a greater angle more of the gravitational force will be directed in the same direction as the motion of the cart ergo increasing the carts acceleration along the ramps allowing the cart to reach a higher speed.

(d)

i.

√   No

With the given equation one would expect that when the mass is doubled the vavg would double but that isn’t the case with the graph. From, 5 kg to 1 kg mass is doubled but not vavg.

ii.

√   No

One would expect that by increasing of the distance between bumps, vavg would also increase however the given equation suggests the opposite, It (equation) suggests that to increase vavg one could shorten of (distance between bumps), but that makes no physical sense because the bumps slow down the cart so one would want to increase the distance between bumps to increase vavg not decrease the distance.

4. Question: (7 points, suggested time 13 minutes)

A circuit contains a battery and four identical resistors arranged as shown in the diagram above.

(a) Rank the magnitude of the potential difference across each resistor from greatest to least. If any resistors have potential differences with the same magnitude, state that explicitly. Briefly explain your reasoning.

Ranking:
Brief explanation:
Resistor B is now removed from the circuit, and there is no connection between the wires that were attached to it. The new circuit diagram is shown below.

(b) When resistor B is removed, does the current through resistor A increase, decrease, or remain the same?
________Increase __________Decrease ___________Remain the same
Briefly explain your reasoning.

(c) When resistor B is removed, does the current through resistor C increase, decrease, or remain the same?
_________ Increase ________ Decrease ___________ Remain the same
Briefly explain your reasoning.

Answer/Explanation

Ans:

(a)

A has same potential difference as D. B \(\frac{1}{3}\) , C are less than A \(\frac{1}{3}\) , D but equal to each other.

Brief explanation:

Since A and D are identical resists with the same amount of current through both of them, they have the same potential difference (V – IR) B and C are in a parallel structure so current gets split between them, making each of their potential differences v = \(\frac{1}{z}\) IR

(b) 

   X    Decrease

Current (I) is              \(\frac{potential difference}{resistance}\)            (I = \(\frac{V}{R}\)  )

When B and C where in a parallel circuit, B \(\frac{1}{3}\)  C’s combined resistance were less than either one of them. Therefore, is A, C, and D are all in a series there will be more resistance which equals less current.

(c)

  X   Increase

When B \(\frac{1}{3}\), C where in a parallel circuit, current was split evenly between them. Current is the same for all resisters in a serves, and since the removal of be creates a series circuit including C, C will get all of the current within the circuit.

5. Question: (7 points, suggested time 13 minutes)

The figure above on the left shows a uniformly thick rope hanging vertically from an oscillator that is turned off. When the oscillator is on and set at a certain frequency, the rope forms the standing wave shown above on the right. P and Q are two points on the rope.

(a) The tension at point P is greater than the tension at point Q. Briefly explain why.

(b) A student hypothesizes that increasing the tension in a rope increases the speed at which waves travel along the rope. In a clear, coherent paragraph-length response that may also contain figures and/or equations, explain why the standing wave shown above supports the student’s hypothesis.

Answer/Explanation

Ans:

(a)

Point P is closer to the oscillator and therefore has a greater mass hanging below it. The force of tension is equal and opposite to the force of gravity, so since Fg = mg and the mass is greater at point P than point Q, the Ft is greater at point P than point Q. FT = – Fg

(b) The wavelength (λ ) at point P is greater than that at point Q. Since V ∝ λ as shown by v = λf, the velocity at point P must also be greater than point Q. Since point p are point Q are or the same string and only their tensions differ, the greater tension at point P  correlates with the greater wavelength and therefore speed.

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