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ME 230 - Dynamics Your Name:_________________
Tutorial 1 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
Rectilinear Motion
This tutorial will introduce the methods of solving straight line kinematics problems by examining two exercises, one from your textbook and the other involving experimentation with a bouncing ball. In addition, it will provide a look ahead at Newton’s Second Law and applications of kinetics.
1. A race car starts from rest and accelerates at [pic] for 10 seconds. The brakes are then applied, and the car has a constant acceleration of -30 ft/sec2 until it comes to rest. Determine (a) the maximum velocity; (b) the total distance traveled; (c) the total time of travel. (Text problem 2.33)
Ask yourself (and answer!) the following questions (by quick inspection) in order to plan solution strategy:
Should problem be broken into two parts or intervals? Where should it be divided?
Does the first interval have either constant acceleration or velocity? How about the second interval? Why does this matter?
At what time will the velocity be maximum?
Does determination of maximum velocity depend on the braking phase?
Will the car ever have a negative acceleration and a positive velocity? Where?
Now execute solution:
i. Derive general equation for v(t) during the acceleration phase from a=dv/dt.
ii. Find vmax at end of acceleration phase.
iii. Derive general equation for s(t) during acceleration phase from v=ds/dt.
iv. Find s at end of acceleration phase.
v. During braking phase, derive general equation for v(t) starting with a=dv/dt.
vi. Find elapsed time during braking phase.
vii. What is total elapsed time during acceleration and braking?
viii. Derive general equation for s(t) during braking phase, starting with v=ds/dt.
ix. Compute total distance traveled by car during acceleration and braking.
2. A rubber ball is dropped to the floor from a height of 4 feet. Measure the height of the ball on rebound with a tape measure by taking the average measurement of 5 or more “first” bounces.
Ask yourself these questions:
Is vertical acceleration constant? Is vertical velocity constant?
Do you know what the vertical acceleration is at every point? Where do you know it?
Determine:
i. The time it takes for the ball to reach the floor (do this analytically)
ii. The velocity of the ball just before it hits the floor
iii. The velocity of the ball just as it leaves the floor on the first bounce.
iv. The total time elapsed from release of the ball (at 4 feet) until it reaches zero velocity at the peak of its first bounce. Verify this experimentally as best you can. Justify any discrepancies.
v. Optional Suppose the ball would have been dropped from rest from a position such that it strikes the floor 0.5 s later and then remains in contact with the floor for 0.05 s before it rebounds in a vertical direction. Draw a graph of the acceleration a(t) versus time t and derive an equation for the velocity as a function of time.
OMIT? Used in the past, but includes is much more advanced than the coverage to date appears to merit.
v. Suppose that the ball weighs 0.08 lb. and it receives an upward force from the floor of 0.5 lb. for 0.15 sec. Use Newton’s Second Law to determine the average acceleration of the ball from the bounce, and the maximum height the ball will reach after the bounce.
ME 230 - Dynamics Your Name:_________________
Tutorial 2 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
2-D Kinematics: Curvilinear Motion and Polar Coordinates
This tutorial will reinforce the methods of solving curvilinear dynamics problems by examining three exercises.
Review of polar coordinate position, velocity and acceleration vectors:
Position: [pic] (1)
[pic]
Because [pic] rotates with angular velocity [pic], [pic]
Velocity: [pic] (2)
[pic]
For positive [pic], [pic]
Acceleration: [pic] (3)
1. A pilot wants to drop supplies to remote locations in the Australian outback. He intends to fly horizontally and release the packages with no vertical velocity. Derive an equation for the horizontal distance d at which he should release the package in terms of the airplane’s velocity v0 and the altitude h (Text problem 2.75).
Ask yourself these questions:
Is horizontal acceleration constant? Is horizontal velocity constant?
Is vertical acceleration constant? Is vertical velocity constant?
What is the horizontal velocity? What is vertical acceleration?
Is this a projectile problem?
Execute solution:
i. In a small sketch, choose a reference frame, label it, and define [pic]. Define vectors from origin of reference frame to some point on the trajectory of the package.
ii. Integrate [pic]to get [pic]; then integrate [pic]to get [pic].
iii. Define the position vector to the target, [pic].in terms of h and d.
iv. Equate [pic] and [pic] and solve for d.
2. Perform an experiment to verify the motion of a projectile. Using either the model catapult or cannon, launch projectiles and measure their maximum height and range. Do this by averaging at least five shots.
|Iteration |Max. Height (in.) |Max. Range (in.) |
|1 | | |
|2 | | |
|3 | | |
|4 | | |
|5 | | |
|AVERAGE | | |
a) Now measure whatever parameters you feel are necessary (by using the protractor, inclinometer, and tape measure), and use your experience from the previous problem to analytically determine the initial velocity of the projectile as it was fired.
b) Analytically determine the expected maximum height and range for a new launch angle that is significantly (10 to 20 degrees) higher. Then test your results by firing the cannon or catapult from that angle . Compare analytical with experimental results. Account for any differences.
ANALYTICAL:
Predicted maximum height: ________________
Predicted range: ________________
EXPERIMENTAL :
|Iteration |Max. Height (in.) |Max. Range (in.) |
|1 | | |
|2 | | |
|3 | | |
|4 | | |
|5 | | |
|AVERAGE | | |
c) Experiment to determine the launch angle that results in the maximum range. Discuss whether this is what you expected, and rationalize any discrepancies.
3. A boat searching for underwater archaeology sites in the Aegean Sea moves at 4 knots and follows the path r = 10( m, where ( is in radians. (A knot is one nautical mile, or 1853 meters per hour). When (=2( rad, a) determine the boat’s velocity in terms of polar and cartesian coordinates (Text problem 2.144)
a) Draw vector diagram depicting boat’s trajectory. Show [pic].
b) Determine [pic], [pic], and [pic] from information given.
c) Write velocity equation (2) from first page in space below. Identify which portion corresponds to vr and which corresponds to v(.
OMIT?? The tutorial is very long if this is included.
d) Write equation (2) in terms of a single unknown, [pic], and solve.
e) Write the velocity, [pic], in polar form.
f) Convert velocity to cartesian form.
ME 230 - Dynamics Your Name:_________________
Tutorial 3 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
Newton’s 2nd Law
This tutorial will examine methods of using Newton’s Second Law in solving kinetics and kinematics problems. A physical model of the first problem will be used to lend real-world significance and to confirm the analytical results.
1) A block resting on a ramp is attached to a string that is pulled up a ramp by a force gauge. We wish to develop a procedure to determine the static and kinetic coefficients of friction. Following that we will use Newton's Second Law and determine the motion of the block when pulled up the ramp by a weight.
i) Draw a FBD of the block on a ramp inclined at an angle (.
ii) Perform a static analysis to symbolically determine the static coefficient of friction between the block and the ramp from the weight of the block W and the tension T applied by the force gauge and the angle (.
iii) Measure the weight of the block and the angle of the ramp and perform a series of experiments to determine the minimum tension necessary to move the block. Compute the static coefficient of friction from your results.
Wblock _________ lb Ramp angle [pic]= _________ degrees
iv) As a check on your result for the static coefficient of friction, compute the minimum tension to move the block on a level surface, measure that tension in a series of experiments, and compute the static coefficient of friction and compare the result with that obtained from ramp angle (.
iv) We now wish to analyze the motion of the block in a dynamic configuration to determine the kinetic coefficient of friction. Repeat the previous series of experiments while pulling the block at a constant speed with a constant tension as determined by the force gauge. (Note the kinetic coefficient of friction must be smaller than the static one.)
v) We now wish to analyze the motion of the block if it were pulled up a ramp by a weight force. Draw three separate FBDs of the block on the ramp, the pulley, and the weight.
vi. Using Newton’s Second Law, determine the acceleration of the block up the ramp if the weight force were 2 lb. Think ..Is the acceleration constant?
[pic]=_______ft/sec2
2) The acceleration of the 20-lb collar A is [pic]ft/sec2. What is the magnitude of the force [pic]? (Text problem 3.53)
i. Draw a FBD of the collar, showing the vector forces acting on it.
ii. Symbolically describe the solution to this problem by using Newton’s Second Law, [pic]. (That is, write the relevant forces, mass, and acceleration in symbolic vector form.) Hint: dot products and unit vectors may be useful here, because you need to resolve the forces and acceleration in a direction parallel to the bar.
iii. Solve for the magnitude of [pic].
3) A 4 lb ball revolves in a horizontal circle as shown. Knowing that L=3 ft. and that the maximum allowable tension in the cord is 10 lb, find a) the maximum allowable speed, and b) the corresponding angle of [pic]. (Hint: solve using normal and tangential components.)
i. Draw a FBD of the ball. It is suggested that you show a “side” view, with the origin of a Cartesian reference frame at the center of the ball.
ii. Ask yourself: What is the tangential acceleration in this case? Why?
iii. Is there a normal acceleration component? Normal acceleration [pic]. Write an expression for [pic].
iv. Using Newton’s Second Law, solve for the maximum allowable speed, v, and for the corresponding angle,[pic].
ME 230 - Dynamics Your Name:_________________
Tutorial 4 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
Work and Energy Methods
Relationships of Interest
Work: [pic]
For weight: [pic] For springs: [pic]
Potential Energy:
For gravity: [pic] For springs: [pic] (S = stretch, [pic])
Kinetic Energy: [pic]
Conservation of Energy: T + V = Constant or T1 + V1=T2 + V2
1) A weight will be dropped from the edge of your table while attached to an elastic sping (rubber band), much like a bungee jumper. Determine a) the maximum length below the suspension point reached by the bottom of the weight, and b) the maximum force (tension) in the spring.
Weight of lead ball:_____________lb. Unstretched spring length:__________in.
Determine the average spring constant k by using the tape measure and the force gauge (take at least five force readings from the unstretched spring length to the maximum elongation length) and average the spring constant results to obtain
k =___________lb/in.
Define the system to be analyzed. Name all the components of the system.
Which of the following forces is not conservative: weight, friction, spring. Explain.
If all forces doing work on the system are conservative (are they?), then the principle of conservation of energy can be used. State the principle symbolically.
What information is necessary to determine V and T?
Draw a diagram showing a reference frame and choose appropriate positions of the weight (1 and 2 etc.) that are necessary for solving the problem.
Apply the principle of conservation of energy to solve for the stretched spring length, s.
Determine the maximum distance below the suspension point that the weight reaches. Experimentally verify your results.
Draw a free body diagram of the weight at maximum spring extension
Calculate the maximum tension in the spring. Verify experimentally.
Qualitatively, where does the maximum velocity occur?
2) If the ball is released from rest in position 1, use conservation of energy to determine the initial angle ( necessary for it to swing to position 2. (Text problem 4.86.)
ME 230 - Dynamics Your Name:_________________
Tutorial 5 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
Impulse, Momentum, Impact, and Conservation of Linear Momentum
Relationships of Interest
Work: [pic]
Potential Energy
For weight: [pic] For springs: [pic] (S = stretch, [pic])
Kinetic Energy: [pic]
Conservation of Energy: T + V = Constant or T1 + V1=T2 + V2
Linear Impulse-Momentum:: [pic]
Conservation of Linear Momentum: [pic] or [pic]
Impact [pic]
Coefficient of restitution: [pic]
Perfectly plastic: e = 0 Perfectly elastic: e = 1
1) A 1 oz. bullet is fired with velocity 1600 ft/sec into block A, which weighs 10 lb. The coefficient of kinetic friction between block A and the cart BC is 0.50. Knowing that the cart weighs 8 lb. and can roll freely (and that 1 lb. = 16 oz.), determine:
a) the final velocity of the cart and block
b) the final position of the block on the cart
This exercise will give you an opportunity to carefully define the system you are working with, and to see that analysis of more than one system is sometimes necessary.
a) If conservation of momentum is to be used, there can be no external forces. What choice of systems would make this possible so that you can easily solve for the final velocity of the cart? Sketch the system three times, first showing initial conditions, then the moment when the bullet strikes the block, and finally showing the system after the block has stopped sliding on the cart.
Using conservation of linear momentum, determine the final velocity of the cart, block, and bullet (when the block no longer slides on the cart).
b) In order to find the find the position of the block on the cart, start by using conservation of linear momentum to determine the velocity of the block immediately after it is impacted by the bullet. Define and sketch the system that would allow this. Is the friction force impulsive?
Use conservation of linear momentum to determine the velocity of the block immediately after impact.
Are forces acting on the block conservative? Can conservation of energy be used to solve for the final velocity of the block?
Draw two FBDs, one of the block with the bullet embedded, and the second of the cart.
Consider the block-bullet system. Using work and energy, solve for the distance moved by the block until it has no relative motion on the cart.
Now consider the cart. Using work and energy, solve for the distance moved by the cart until there is no relative velocity between the block and the cart.
Determine the distance moved by the block relative to the cart as it slides on the surface of the cart.
2) A lead ball weighing _____ lb. hanging from a string of length l = ____ in. is released from rest at an angle of (A = 20 degrees with the vertical. As it swings past vertical, it strikes a block weighing ____ lb., which slides a distance d = _____ in., and then the ball continues to swing through an angle of ____ degrees. After we determine in class the kinetic coefficient of friction between the block and the ground, _____, we want to determine the coefficient of restitution for the collision as our final objective. To do this we would like to have the ball swing freely after the impact knocks the block out of the way. (Note: measurements missing above will be made during class)
Using work and energy, symbolically determine the velocity of the ball just before striking the block and just after collision. Are the forces conservative? Can you use conservation of energy to do this?
Using work and energy, symbolically determine the velocity of the block just after the collision. Are the forces conservative? Can you use conservation of energy?
Using your equation and your experimental results, determine the velocity just before and after the collision of the ball and the block.
Determine the coefficient of restitution for the collision.
EXTRA MATERIAL NOT USED:
Using e calculated above, increase the pull-back angle to at least 30 degrees, release the ball to strike the block, and measure the rebound angle of the ball after impact. Predict the distance moved by the block and compare with an actual measurement.
d predicted ______ d measured _______
ME 230 - Dynamics Your Name:_________________
Tutorial 6 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
Angular Momentum and 2-D Kinematics
Relationships of Interest
Angular Momentum: [pic] and [pic]
Rate of Change of Angular Momentum [pic]
Conservation of Angular Momentum: [pic] or [pic]
1) The mechanical governor shown consists of two spheres of identical masses m mounted on two rigid rods of negligible mass and of length L, rotating freely about a vertical shaft. An internal mechanism adjusts the angular position ( of the rods to secure the desired angular velocity of the shaft. If the angular velocity of the spheres is given as (o when the rods are horizontal ((o = 90o), determine its magnitude for any angle ( in terms of (o , then find its magnitude when ( = 45o.
Draw a coordinate reference frame on the diagram above, in cylindrical coordinates, with the origin at O.
Is angular momentum conserved about the z axis? Why? Write the symbolic statement of conservation of angular momentum about the z axis.
Write an expression for the angular momentum Ho at (o = 90o in terms of m, L and (o.
Write a similar expression the angular momentum Ho at any other angle ( in terms of m, L, (, and (.
Equate the expressions for angular momentum at 90o and at any angle (. Why can you do this? Obtain a general expression for the angular velocity at any position ( in terms of the angular velocity at (o (when ( = 90o).
From the above expression, what is the angular velocity of the shaft at ( =45o.
2) The cylinder of diameter R rolls without slipping on the plane surface. Point A is moving to the right a constant speed vA. What is the angular velocity vector of the cylinder. Find the velocity and acceleration of points B, C D, and E. (similar to problems 6.27 and 6.85)
What is the angular velocity vector for the disk?
Write the position vector symbolically for points B, C, D, and E in terms of rA = R j, where R is the cylinder radius, in terms of R and (.
Take the time derivative of each position vector to find the velocity vector of each point.
Roll the cylinder across the table, noting the path described by a point on its rim. In the space below, trace out the path described by that point and label the points corresponding to B, C, D, and E. Draw small arrows representing the velocity of each of the four points.
Take the time derivative of each velocity vector to find the acceleration vector of each point.
ME 230 - Dynamics Your Name:_________________
Tutorial 7 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
2-D Kinematics and Mass Moment of Inertia
1) At the instant shown, the length of the boom is being decreased at the constant rate of 6 in/sec and the boom is being lowered at the constant rate of 0.075 rad/sec. Knowing that ( = 30o, determine: a) the velocity of point B, and b) the acceleration of point B.
Choose and indicate appropriate reference frame(s).
Write the position vector [pic]for the boom tip in terms of symbols you define for the coordinate system you select.
Symbolically differentiate the position vector [pic] to determine the velocity vector [pic]for the boom tip.
Symbolically differentiate the velocity vector [pic]to determine the acceleration [pic]for the boom tip.
Numerically evaluate your equations to determine the velocity vector [pic] and the acceleration vector [pic]for the boom tip at the instant shown.
2) A cylindrical shell (pipe) and a solid cylinder, each of mass m and radius R, are released from rest on a surface inclined at angle ( and allowed to roll a distance of D feet. Determine the time (in seconds) it takes for each to roll this distance.
Write two general equations for the translational and rotational motion.
If there is no slipping, what equations exist that relate the angular displacement, velocity, and acceleration to the translational displacement, velocity, and acceleration.
Construct a free body diagram for the system.
Write the general equations for the angular acceleration of the pipe and cylinder in terms of the appropriate moments of inertia I and the other constants and variables.
Solve for the acceleration of the center of mass, a, for both the pipe and the cylinder by using appropriate equations for I.
From the above equations, and the experiment that you have set up, determine the time for each object to roll a proper distance down the incline.
Experimentally verify your results. Are your results dependent upon the mass of the rolling objects?
ME 230 - Dynamics Your Name:_________________
Tutorial 8 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
Review: Work and Energy Methods, 2-D Kinematics
1) The objective of this experiment is to test the principle of conservation of energy if a ball is released from rest in position 1 and swings to position 2 after the cord partially wraps around a small diameter rod that contributes negligible friction. You will first analytically determine the initial angle ( necessary for it to swing to position 2 and then experimentally verify your results.
a) Define the system to be analyzed by sketching the system, showing the initial and final positions of the ball.
Draw a free body diagram of the ball. What force(s) are acting on the ball? What forces are doing work and are those forces conservative?
Determine angle of release, (, so the ball will reach the desired final, horizontal position.
Experimentally verify your results using the test fixture and a protractor. ( = _______.
b) Now consider the angle ( to enable the ball to rotate around the rod with a minimal velocity while the cord still remains straight. First draw a free body diagram for the configuration when the ball is at its apex.
Determine the ball's tangential velocity at the apex of its motion using Newton's second law in the radial direction.
Use the principle of work-energy to estimate the angle ( necessary to enable the motion that you determined at the apex.
Experimentally verify your results using the test fixture and a protractor. ( = _______.
2) A roll of paper of 72 mm radius is initially at rest on a horizontal surface. A horizontal force P applied to the paper produces the following accelerations: [pic]= 300 mm/sec2 and [pic]= 780 mm/sec2, both to the right. From the equation relating the accelerations at points A and B, determine a point on the vertical surface of the roll of paper which a) initially has no acceleration and b) initially has an acceleration of 500 mm/sec2 to the right.
ME 230 - Dynamics Your Name:_________________
Tutorial 9 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
Energy and Momentum Methods for Rigid Body Motion
Relationships of Interest:
For general planar motion: [pic]
Work and Energy: [pic]
Conservation of Energy: [pic]
I for Slender Rod: About C.M., [pic] About end, [pic]
Potential Energy: For gravity, [pic], For spring, [pic] (S = stretch, [pic])
Linear Impulse and Momentum: [pic]
Angular Impulse and Momentum: [pic], where [pic]
1) A 4-ft. slender bar weighs 40 lb. with mass center at B and is released from rest in the position for which [pic] is essentially zero. Point B is confined to move in the smooth vertical guide, while end A moves in the smooth horizontal guide and compresses the spring as the bar falls. Determine a) the angular velocity of the bar as position [pic] = 30o is passed and b) the velocity with which B strikes the horizontal surface if the stiffness of the spring is 30 lb./in.
Part a)
i. Are the forces conservative? If so, write the symbolic statement of conservation of energy. Then write the symbolic statement of kinetic energy for a rigid body. Is the symbolic statement the same if point A is used as the reference point rather than the center of mass?
ii. For the first 30 degrees of motion, is the spring involved?
iii. Determine the kinematic relation between the velocity [pic] of the center of mass and the angular velocity [pic] of the bar. Do this by finding the instantaneous center and recalling that [pic].
iv. Find the kinetic energy of the system at [pic] = 30o ,and using conservation of energy, solve for the angular velocity of the bar at that point.
Part b)
v. Find the kinetic energy for the rod when it reaches the horizontal position.
vi. Now consider the motion of the rod. Find the potential energy for the rod at the initial position and the final position when the rod strikes the horizontal surface.
vii. Using conservation of energy, find the velocity with which B strikes the horizontal surface
2) A 30 g bullet is fired with a horizontal velocity of 400 m/s into a 4 kg wooden beam AB suspended from a pin support at A. Knowing that h = 600 mm and that the beam is initially at rest, determine a) the velocity of the mass center G of the beam immediately after the bullet becomes embedded, and b) the reactions at A, assuming that the bullet becomes embedded in 1 ms.
i. Define your system to be the rod and the bullet. Draw a free body/momentum diagram of the system to illustrate the linear and angular impulse-momentum equations for an impulse equal to the change in momentum. The diagram should include a figure representing the period during which the bullet becomes embedded (showing the external forces on the rod), and figures showing the system just after and just before the bullet becomes embedded.
ii. Write expressions for angular impulse and momentum and linear impulse and momentum pertaining to the three cases above. You may neglect the mass of the bullet.
iii. Determine the kinematic relationship between the velocity of the center of mass, [pic], and the angular velocity of the bar, [pic].
iv. From these three relationships, solve for [pic], [pic], and the reaction(s) at A. Does the reaction at A in the vertical direction change because of the impact?
ME 230 - Dynamics Your Name:_________________
Tutorial 10 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
Momentum Methods for Rigid Body Motion and Vibrations
Relationships of Interest:
Rigid Body Motion:
Linear Impulse and Momentum: [pic]
Angular Impulse and Momentum: [pic], where [pic] (about C..M.)
I for Sphere: About C.M., [pic]
1) The 220 mm diameter bowling ball show has a mass of 7.25 kg. The instant the ball comes in contact with the alley, it has a forward velocity v of 7 m/sec and a back spin [pic]of 6 rad/sec. If the kinetic coefficient of friction between the ball and the alley is 0.15, determine:
a) the elapsed time before the ball begins to roll without slipping , and
b) the distance traveled at this point.
i) Illustrate the motion of the ball for the equation: System External Impulses1-2 = System Momenta2 - System Momenta1
ii) Label a convention for positive x , y, and moments. Write the two symbolic equations for rigid body motion (linear and angular). Apply these equations to the appropriate information from the above diagrams.
iii) Determine the kinematic relation between v and [pic]. At what point in the ball’s trajectory would it make sense to do this?
iv) Combine the kinetics and kinematics relationships to solve for the time elapsed until ball starts rolling without slipping.
v) From kinematics, determine the corresponding distance traveled.
ME 230 - Dynamics Your Name:_________________
Tutorial 11 Section No.:_________________
Partners:_________________ _________________ _________________
_________________ _________________ _________________
Vibrations and Three-Dimensional Rigid Body Kinematics
Relationships of Interest:
Vibrations:
Period [pic] (sec) Natural Frequency f =[pic] (cps)
Circular Natural Frequency [pic] rad/sec
Three-Dimensional Rigid Body Kinematics:
Velocity:
Acceleration:
1) The single degree of freedom spring-mass system symbolically shown below has been provided in the classroom. The spring constant k is measured in class to be ________. The lead balls weigh _______ lb each. Determine the quantities described below.
i) Draw a free body diagram of the mass. Using Newton’s Second Law, write the equation of motion for the system.
ii) Analytically determine circular natural frequency, the period of oscillation, and the natural frequency using the classroom data.
iii) Experimentally determine period and natural frequency of the system. Compare the circular natural frequency with your analytical results.
iv) Now predict the natural frequency for this system with two (equal) weights instead of one. Experimentally verify your results.
2) The gyroscope shown has a primary reference frame that is stationary and a secondary reference frame that rotates about the y-axis with a constant angular velocity (y. The wheel spins about the z-axis with a constant angular velocity (z. Because (z is difficult to measure without the proper equipment (e.g., a strobe light), we will assume that (z = 180 rad/s and (y = 25 rad/s.
For the gyroscope, the measured value of rA/O is 1.5 in and the measured value of the disk radius, rB/O = R, is 1.25 in. (This is like Problem 9.16 from lecture.)
2.a) Complete the equations ( = ________ rad/s and (rel = _________ rad/s.
2.b) Using the symbols rA/O, R, and (. write the symbolic equations with which you can compute aA and aB.
2.c) For your experiment, numerically evaluate the average aA and, for ( = 30o and 90o, the average aB.
-----------------------
vA = v( + ( ( rA/B Eq. (9.1)
for ( = ( + (rel Eq. (9.5)
aA = aB + ( ( rA/B + ( ( (( ( rA/B) Eq. (9.2)
for ( = d( x/dt i + d( y/dt j +d( z/dt k + ( ( ( Eq. (9.4)
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