081124

How was the system set up?

What data did you take?

What was the uncertainty of each measurement?

What results did you get from the data?

What is your answer to the main question and what is the uncertainty in your answer?

Inclines

• Acceleration vs. slope

What is the rate of change of the acceleration of a ball on the grooved track with respect to the slope of the track?

• Uniformity of acceleration

Is acceleration on a grooved track at a constant slope independent of its velocity and position on the track?

• coefficient of rolling friction

What is the coefficient of rolling friction for a steel ball on a grooved steel track?

• analysis by vectors

Is the acceleration of a steel ball on a grooved steel track, at a given small slope, equal to the acceleration that would result from a net force equal to the component of the ball's weight in the direction of motion?

• two tracks

If a ball released from rest at some point on one incline rolls freely onto to another incline, is its change in velocity as it then rolls the length of the second incline dependent or independent of the position at which it was released on the first incline?

• conservation of energy

If a ball rolls without slipping up an inclined grooved track, comes to rest then rolls back down, under the influence of only gravity, normal force and friction, does it end up back at its starting point with its original kinetic energy, how much potential energy does it gain, and how much energy is lost to friction as it travels in each direction?

 

Is the sliding distance of a block proportional to the area under the force vs. length graph of the rubber band chain used to project it along the tabletop?

 

(remember detection of slipping)

Rubber bands

• What is the nature of the graph of tension force vs. length for a given rubber band?

• How is the area beneath a segment of the graph determined and what does it mean?

• How is the slope of the graph at a given point determined and what does it mean?

• If the rubber band is stretched out between two given points in a coordinate plane, how do we determine from the coordinates of the points and the tension vs. length graph the magnitude and direction of the tension force acting on a given endpoint?

Pendulum

• period vs. length

What is the period vs. length function for a hand-held simple pendulum consisting of a washer supported by a thread, and oscillating with amplitude not exceeding 10% of its length?  Does the period depend on the mass of the washer?

• force vs. pullback

How does the horizontal force necessary to hold a pendulum of given mass and length at a point away from its equilibrium position depend on the displacement of that point from the equilibrium position?

• analysis by vectors

How does the observed force required to hold a pendulum of given mass and length at a point away from its equilibrium position compare with the force predicted by analyzing the tension in the pendulum string?

Atwood machine

• light pulley, acceleration vs. relative mass

On an Atwood machine originally consisting of equal masses m suspended from a pulley with negligible mass and friction, if mass `dm is transferred from one side to the other, what is the resulting acceleration of the system?

 

What are the limits on the coefficient of friction of this system?

• massive pulley

What are the limits on the moment of inertia of the pulley in the preceding?

• combined with incline

How much mass must be transferred from one end to the other if one of the masses is on an incline at angle theta to horizontal, subject to sliding friction with coefficient mu, in order to achieve a given acceleration (up or down the incline)?

Rubber bands in equilibrium

• Condition of equilibrium

If a paperclip is held in equilibrium on a horizontal surface by three rubber bands stretched in three different directions, how nearly do the predictions of the force vs. length graphs and the analysis of the vector forces confirm the equilibrium?

 

Does it matter how the coordinate axes are oriented?

Rotating Strap

• The standard system here consists of a metal strap which is constrained to rotate on top of a smooth die. It is easy to give the system a spin and estimate to within 10 degrees or so the angle through which it rotates, and the time required to complete the rotations.  Take measurements sufficient to determine whether and if so how the angular acceleration of the 'coasting' strap is depends on the angular velocity of the strap.

• Using a chain of rubber bands, apply a known torque through a small angular displacement to a metal strap which is constrained to rotate on a smooth die.  Time the strap as it comes to rest, and use your results to determine, as accurately as you can, the moment of inertia of the strap.  Assuming that the moment of inertia of the strap is well approximated by the moment of inertia of a thin rod of the same length and mass, determine the mass of the strap.  (231 determine to within +-1% the difference between the moment of inertia of the strap and that of the rod).

Repeat with a stack of dominoes on each end of the strap.

 

Incidentally estimate the coefficient of friction between the strap and the die.

 

(80 g (4 dominoes balance strap on second strap fulcrum 1 domino), 30 cm length, .08 kg * (.30 m)^2 / 12 = .0006 kg m^2; what is limit of precision using said balance?)

• By 'shooting' the rotating strap with a sufficiently massive rubber band, it is possible to get the system to rotate for a couple of seconds.  By 'shooting' the rubber band across the room or down the hall, it is possible to estimate its velocity just after release. 

(point of reference:  a 1-gram rubber band pulled back 1 cm with a 1-N average force will have an 'ideal' PE of .01 J; if totally converted into KE the velocity will therefore be about sqrt(20) m/s = 4.4 m/s.  Horizontal range from height 1 meter should be about 2 meters.)

(measure with and against a breeze to get an idea of how much to correct for wind resistance)

(video is also possible)

• (does rubber band lose range if shot repeatedly?)

Projectiles

• The horizontal range of a horizontal projectile can be used as a measure of horizontal velocity

If an object is projected from a known height into free fall with zero vertical velocity, then if air resistance can be neglected its horizontal velocity remains constant and its time of fall is easily determined by the distance of fall and the acceleration of gravity.  If the horizontal range is measured, it is therefore easy to find the horizontal velocity.

• The horizontal range of a projectile projected in a know direction can be used to determine its initial velocity.

If an object is projected from a known height into free fall at a known angle, then if air resistance can be neglected its horizontal range can be used to find its initial velocity.

• range vs. slope, range vs. distance down incline

If a ball is rolled down a fixed distance down an incline and allowed to project off the fixed end of the incline into free fall, then the slope of the ramp determines the horizontal range of the projectile and the velocity of the ball on the ramp can be determined from the horizontal range.

 

There is a slope at which the horizontal range is greatest.  A graph of horizontal range vs. slope will indicate the approximate slope at which the maximum range occurs.

 

If the ball rolls without slipping it attains less velocity than if it rolls and slips.  Analysis of ideal horizontal range vs. slope, assuming no slipping, will yield a basis for comparison with actual data.  The comparison should reveal the angle at which slipping begins.

 

For very small values of the slope, when the horizontal range should be negligible, an 'edge effect' causes a significant horizontal range.  The edge effect occurs after the center of the ball reaches the end of the ramp while some point of the falling ball is in contact with the ramp.  This effect imparts a horizontal velocity that causes a significant deviation from the behavior expected if the edge effect is neglected.  Comparison of actual data with the results of an analysis that does not account for the edge effect will clearly show the approximate range of slopes for which the edge effect is significant.

• optimize range from given height

Strap in rotational equilibrium

• What masses are located at what distances from the fulcrum, and what is the sum of their torques?