2.003J/1.053J Dynamics and Control I

Massachusetts Institute of TechnologyDepartment of Mechanical Engineering

Fall 2010, Problem Set 3Issued: September 23, 2010

Online Survey due: September 25, 2010, 11:59 PMProblem set due: September 30, 2010, 9:30 AM

The online portion of the homework (the online survey) is to be completed by before the end of Saturday.It accounts for 25% of the problem set grade. Survey questions are included below for convenience but mustbe answered online.

The problem set solutions are due at lecture on Thursday. At lecture there will be a box for each recitation.Put your solution into the box for your recitation. It will be returned to you at recitation on Thursday orFriday.

Note: The Thursday 3:00 pm recitation group will be responsible for preparing the nal solutions for thisset due on October 7th.

1. An internal mechanism is used to maintain a constant angular rate Ω = 0.05 rad/s about the z-axis ofthe spacecraft as the telescopic booms are extended at a constant rate. The length l is varied from zeroto 3 m. The maximum acceleration to which the sensitve experiment modules P may be subjected is0.011 m/s2. Determine the maximum allowable boom extension rate l.

Survey Question: Does the operation of the internal mechanism change the total angular momentumof the satellite, computed with respect to the center of mass of the satellite? (a) Yes (b) No

1

2. Problem 15-28: The winch delivers a horizontal towing force F to its cable at A which varies asshown in the graph. Determine the speed of the 80-kg bucket when t=24 s. Originally the bucket isreleased from rest.

Survey Question: Do you expect the speed of the bucket to be constant while the force is constantduring the interval 0 to 12 seconds? (a) Yes (b) No (c) I don't know what principle to apply

3. Problem 15-42: The 75-kg boy leaps o cart A with a horizontal velocity of v′ = 3 m/s measuredrelative to the cart. Determine the velocity of cart A just after the jump. If he then lands on cart Bwith the same velocity that he left cart A, determine the velocity of cart B just after he lands on it.Carts A and B have the same mass of 50 kg and are originally at rest.

Survey Question: While the boy is airborne, the relative velocity between the boy and the rst cart is3 m/s. After the boy lands on the second cart, how will the relative velocity between the boy and therst cart change? (a) Increase (b) Decrease (c) I don't know what principle to apply

2

4. Problem 15-48: The free-rolling ramp has a weight of 120 lb. If the 80-lb crate is released from restat A, determine the distance the ramp moves when the crate slides 15 ft down the ramp to the bottomB.

Survey Question: Does the outcome of this question depend on the coecient of friction between theweight and the rolling ramp. Assume that, if needed, the sliding weight is brought to rest by a bumperat the end of the ramp. The outcome is: (a) Independent of friction (b) Dependent on Friction (c) Idon't know what principle to apply

5. Problem 15-91 (modied):

(a) Find an expression for the angular momentum of the car with respect to a point on the axis ofrotation where the rod attaches to the pivot. Treat the car as a particle.

(b) At t=5 seconds the external driving forces and moments are turned o leaving the car to coast atconstant rotation rate. Find an expression for the linear momentum, P , of the car.

(c) Compute the time derivative of P with respect to the Oxyz inertial frame. By Newton's 2ndlawthe result must be the force applied to the car. Explain the physical meaning of the force thatyou have found.

Survey Question: What do you expect dPdt to yield: (a) 0 N (b) Mg (c) a tangential force (d) a radial

force

3

6. A 3 kg monkey runs up the shaft shown in the gure. The shaft rotates at a constant rate of ω=2rad/s. The speed of the monkey running along the shaft is 1.5 m/s.

(a) Find expressions for the velocity and acceleration of the monkey with respect to a xed inertialframe Oxyz located at the point the inclined shaft connects to the vertical rotating shaft.

(b) Find an expression for the angular momentum of the monkey with respect to the origin at O.What is the direction of the angular momentum vector? Draw an arrow that begins at O torepresent the angular momentum vector, h/O.

(c) Compute the time rate of change of h/O with respect to the Oxyz frame. It should have two terms.Give a physical interpretation of the meaning of each term.

m = 3 kg

Oxyz 4 m

3 m

r

ω

z, k

^

^

Survey Question: In what direction is the angular momentum vector found in part (c).

(a) Z direction (b) perpendicular to the shaft that supports the monkey (c) parallel to the shaftsupporting the monkey

4