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Question: (7 points, suggested time 13 minutes)

Identical blocks 1 and 2 are placed on a horizontal surface at points A and E, respectively, as shown. The surface is frictionless except for the region between points C and D, where the surface is rough. Beginning at time At , block 1 is pushed with a constant horizontal force from point A to point B by a mechanical plunger. Upon reaching point B, block 1 loses contact with the plunger and continues moving to the right along the horizontal surface toward block 2. Block 1 collides with and sticks to block 2 at point E, after which the two-block system continues moving across the surface, eventually passing point F.
(a) On the axes below, sketch the speed of the center of mass of the two-block system as a function of time, from time tA until the blocks pass point F at time tF . The times at which block 1 reaches points A through F are indicated on the time axis.

(b) The plunger is returned to its original position, and both blocks are removed. A uniform solid sphere is placed at point A, as shown. The sphere is pushed by the plunger from point A to point B with a constant horizontal force that is directed toward the sphere’s center of mass. The sphere loses contact with the plunger at point B and continues moving across the horizontal surface toward point E. In which interval(s), if any, does the sphere’s angular momentum about its center of mass change? Check all that apply.
____ A to B ____ B to C ____ C to D ____ D to E _____ None
Briefly explain your reasoning.

Answer/Explanation

Ans:

(a) 

(b)

__√___ C to D

A change in angular momentum means that the sphere has a torque being applied to it over a period of time, (L = tot), a torque is not applied from A to B because the force is directed towards the center of mass, since the surface is frictionless at B to C and D to E, there is no force that is applied to the sphere, and therefore no torque A torque is only applied during the interval from C to D, since the surface is rough, which means that friction exist. The force of friction applied a torque to the sphere causing the angular momentum to change at C to D.

Question:(12 points, suggested time 25 minutes)

This problem explores how the relative masses of two blocks affect the acceleration of the blocks. Block A, of mass m A, rests on a horizontal tabletop. There is negligible friction between block A and the tabletop. Block B, of mass mB , hangs from a light string that runs over a pulley and attaches to block A, as shown above. The pulley has negligible mass and spins with negligible friction about its axle. The blocks are released from rest.
(a)
i. Suppose the mass of block A is much greater than the mass of block B. Estimate the magnitude of the acceleration of the blocks after release.

Briefly explain your reasoning without deriving or using equations.

ii. Now suppose the mass of block A is much less than the mass of block B. Estimate the magnitude of the acceleration of the blocks after release.

Briefly explain your reasoning without deriving or using equations.
(b) Now suppose neither block’s mass is much greater than the other, but that they are not necessarily equal. The dots below represent block A and block B, as indicated by the labels. On each dot, draw and label the forces (not components) exerted on that block after release. Represent each force by a distinct arrow starting on, and pointing away from, the dot.

(c) Derive an equation for the acceleration of the blocks after release in terms of mA , mB , and physical constants, as appropriate. If you need to draw anything other than what you have shown in part (b) to assist in your solution, use the space below. Do NOT add anything to the figure in part (b).
(d) Consider the scenario from part (a)(ii), where the mass of block A is much less than the mass of block B. Does your equation for the acceleration of the blocks from part (c) agree with your reasoning in part (a)(ii) ?
____ Yes ____ No
Briefly explain your reasoning by addressing why, according to your equation, the acceleration becomes (or approaches) a certain value when m A is much less than mB .
(e) While the blocks are accelerating, the tension in the vertical portion of the string is T1. Next, the pulley of negligible mass is replaced with a second pulley whose mass is not negligible. When the blocks are accelerating in this scenario, the tension in the vertical portion of the string is T2. How do the two tensions compare to each other?
____ T 2 > T1 ____ T2 = T 1  ____ T2 < T 1 
Briefly explain your reasoning.

Answer/Explanation

Ans:

(a) (i)

The magnitude of acceleration would be near Om/s2

Imagine a human pulling a mountain w/a rope. The mass of block a is so much greater that the weight of block D is not sufficient enough to make a movement. No movement = No change in velocity = No acceleration.

(ii)

9.8 m/s2

Block B would pretty man be droping due to its weight and gravity pulling it down, that  would be the “only” force present (Tensibal, there, but negligible if Mblock A < M block B by a huge amount.

(c)

A : Fres = (Tension) = m1a

B : Fres  = (mB g – Tension) = m2a

mBg – mA a = mB a

mBg = mA a + mB a

mBg = a (mA + mB)

a = mBg  / mA + mB

(d)

___√___ Yes        __________ No

As you cm see from the work, if mA

A = \(\frac{m_{B}g}{m_{A}+m_{B}}\)  = \(\frac{9000 g}{9000.01}, \pm \cap \); 999 g ≈ 9.8; is incredibly small, the equation pretty mark becomes  \(\frac{m_{B}}{m_{B}}g\)   which equals 1g.

If mA = 0.01                mD = 9000

(e)

__√_ T2 < T1

If the mass of the pulley is not negligible, energy is required to turn it, meaning acc. Decreases from original. If acceleration decreases, tension decreases as well due to constant mass of Block A. So, T2 is less then T1.

Block             Tension       =  m1a

A                        ↓                        ↓

Question: (12 points, suggested time 25 minutes)

A projectile launcher consists of a spring with an attached plate, as shown in Figure 1. When the spring is compressed, the plate can be held in place by a pin at any of three positions A, B, or C. For example, Figure 2 shows a steel sphere placed against the plate, which is held in place by a pin at position C. The sphere is launched upon release of the pin. A student hypothesizes that the spring constant of the spring inside the launcher has the same value for different compression distances.
(a) The student plans to test the hypothesis by launching the sphere using the launcher.
i. State a basic physics principle or law the student could use in designing an experiment to test the hypothesis.
ii. Using the principle or law stated in part (a)(i), determine an expression for the spring constant in terms of quantities that can be obtained from measurements made with equipment usually found in a school physics laboratory.
(b) Design an experimental procedure to test the hypothesis in which the student uses the launcher to launch the sphere. Assume equipment usually found in a school physics laboratory is available.
In the table below, list the quantities and associated symbols that would be measured in your experiment. Also list the equipment that would be used to measure each quantity. You do not need to fill in every row. If you need additional rows, you may add them to the space just below the table.

(b) Describe the overall procedure to be used to test the hypothesis that the spring constant of the spring inside the launcher has the same value for different compression distances, referring to the table. Provide enough detail so that another student could replicate the experiment, including any steps necessary to reduce experimental uncertainty. As needed, use the symbols defined in the table and/or include a simple diagram of the setup.
(c) Describe how the experimental data could be analyzed to confirm or disconfirm the hypothesis that the spring constant of the spring inside the launcher has the same value for different compression distances.
(d) Another student uses the launcher to consecutively launch several spheres that have the same diameter but different masses, one after another. Each sphere is launched from position A. Consider each sphere’s launch speed, which is the speed of the sphere at the instant it loses contact with the plate. On the axes below, sketch a graph of launch speed as a function of sphere mass.

Answer/Explanation

Conservation of Energy

\(\frac{1}{2}kx^{2} = mgh\)

\(k = \frac{2mgh}{x}\)

(b)

  1. First, use the ruler to measure the compression distance for the spring at each position (A,B,C) – this will be X. Also, measure the mass of the sphere, this will be m.
  2. Next, compress the spring to a chosen position, piece the pin, and load the sphere.
  3. Release the pin, and measure the maximum distance (h) that the ball travelling above its release point.
  4. Repeat the experiment (starting at step 2) multiple times at each position, and repeat it at each of the three positions.

Before ReleaseAfter Release

               

(c) First, the experiment should be repeats multiple times at each position to limit experimenter uncertainty. At each distance calculate the spring constant with the expression \(K = \frac{2mgh}{x} (ii),\) with the average height at that position as h, that measures mass of the sphere as M, and the measured very close at each of the positions, then the hypothesis is confirms, otherwise,  it is disconfirmed.

(d) 

\(\frac{1}{2} kx^{2} = \frac{1}{2}mv^{2}\)

\(kx^{2} = mv^{2}\)

\(v^{2} = \frac{kx^{2}}{m}\)

\(v = \sqrt{\frac{kx^{2}}{m}}\) constant

\(v \alpha \frac{1}{\sqrt{m}}\) v

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

A motor is a device that when connected to a battery converts electrical energy into mechanical energy. The motor shown above is used to lift a block of mass M at constant speed from the ground to a height H above the ground in a time interval Δt. The motor has constant resistance and is connected in series with a resistor of resistance R1 and a battery.
Mechanical power, the rate at which mechanical work is done on the block, increases if the potential difference (voltage drop) between the two terminals of the motor increases.
(a) Determine an expression for the mechanical power in terms of M, H, Δt, and physical constants, as appropriate.
(b) Without M or H being changed, the time interval Δt can be decreased by adding one resistor of
resistance R 2 , where R  2 > R 1, to the circuit shown above. How should the resistor of resistance R 2 be added to the circuit to decrease Δt ?
___ In parallel with            ___ In parallel            ___ In parallel with          ___ In series with the battery,
        the battery                              with R1                           the motor                            R1, and the motor
In a clear, coherent, paragraph-length response that may also contain figures and/or equations, justify why your selection would decrease Δt.

Answer/Explanation

Ans:

(a)   Mechanical power  =  \(\frac{M_{g}H}{\Delta t}\)                  Work = Fd

(b)  _√__ In parallel 
                with R1

If R2 is added in parallel with R, the total resistance of the circuit, excluding the motor, would be less than R1 because the total resistance of a parallel circuit is less than the smallest resistor. According to kirchoff’s loop rule, the voltage drop across an entire circuit must equal zero. When R2 is added in parallel with R1, the voltage drop across that section decreases, which means the voltage drop across the motor most increase assuming the battery stays the same. Mechanical power increases because of this, so time most decrease assuming M and H are held constant.

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

A tuning fork vibrating at 512 Hz is held near one end of a tube of length L that is open at both ends, as shown above. The column of air in the tube resonates at its fundamental frequency. The speed of sound in air is 340 m/ s.
(a) Calculate the length L of the tube.
(b) The column of air in the tube is still resonating at its fundamental frequency. On the axes below, sketch a graph of the maximum speed of air molecules as they oscillate in the tube, as a function of position x, from x = 0 (left end of tube) to x = L (right end of tube). (Ignore random thermal motion of the air molecules.)

(c) The right end of the tube is now capped shut, and the tube is placed in a chamber that is filled with another gas in which the speed of sound is 1005 m s. Calculate the new fundamental frequency of the tube.

Answer/Explanation

Ans:

(a) 

\(v = \frac{1}{2}\lambda \)

λ = 2L                       v =  λf                          v = 2Lf

\(\frac{v}{2f} = L = \frac{3f0}{512.2} = \)

0.33 m

(b) 

(c) 

     \(\frac{1}{4}\lambda = L\)                        λ  = AL

vf= λf                    v = ALf

 \(\frac{\sqrt{}}{4L} = f = \frac{1005}{4.(0.33)} = 761.36 H_{2}\)

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