1.053J/2.003J Spring 2011: Dynamics and Control I Homework 05Massachusetts Institute of Technology Survey Due: October 11th, 2011, 6:00 pmDepartment of Mechanical Engineering Problems Due: October 20th, 2011, 6:00 pm

Online survey questions must be completed by 6pm Tuesday night (October 11th, 2011) and the writtenportion must be completed by 6pm the following Thursday (October 20th, 2011). A link to respond to thesurvey questions will be emailed to each student individually. The written portion must be submitted asa PDF on Stellar. Physical copies will not be accepted. Scanning facilities are available free-of-charge atmost MIT dorms, all MIT libraries, and CopyTech.Discussing assigned written problems with fellow students is encouraged, as this is a great way to gain abetter understanding of the material. However, please attempt the problems individually before doing so.The work that you submit should reflect your own understanding of the problems and should not be copied.Please write down the names of your collaborators on the top of your homework.

Assigned readings: Chapters 6.3 and 6.4 from Williams textbook

(available on Stellar)

Problem 1

Use symmetry rules to find the principal axes that pass through the center of mass for the rigid bodiesshown in the figures (a) to (d) below. A 3D rendering and a 2D cross section are provided for each rigidbody.You can print out these pages and directly label/draw on the figures to complete this problem.

a) An ‘X’ shape

b) An ‘V’ shape, which is half of the previous ‘X’ shape

c) A star shape

d) A bottle shape

Problem 2

A pendulum consists of a small mass m attached at the end of a massless rigid arm of length L. Thependulum is attached (at a frictionless pivot in A) to a block of mass M that rolls in the OX direction ona table. The block is connected to a fixed wall by a spring of constant k and unstretched length l0. Theposition of the block in the inertial frame is XÎ.

Figure 1: Figure for Problem 2

a) Draw the free body diagrams.

b) Find the equations of motion of the system. (Note: you don’t need to solve them)

c) Online survey question: How many independent coordinates are required to completely describe themotion of the system?

(a) 1 (b) 2 (c) 3 (d) 4

Problem 3

A block of mass M is constrained by rollers to motion in the OX direction. A small mass m is attachedat the end of a massless rigid arm. The other end of the arm is attached at A (A is fixed to the block)where a rotor constrains the angle to be θ(t) = ωt. e is the length from A to B. The position of the cartin the inertial frame is XÎ.

Figure 2: Figure for Problem 3

In Problem Set 3 you found the forces the rod places on the mass m:

~Frod/m =[−meθ̇2 cos θ +mẌ

]Î +

[−meθ̇2 sin θ +mg

]Ĵ

a) Find an equation of motion of M by using Newton’s third law to express the forces of the rod on M .

b) Online survey question: Given that ω = 1 rad/s, how many additional independent coordinates arerequired to completely describe the motion of the system?

(a) 1 (b) 2 (c) 3 (d) 4

Problem 4

This is a modified version of Problem 3 from the quiz.Two identical masses are attached to the end of massless rigid arms as shown in the figure. The verticalportion of the rod is held in place by bearings that allow it to rotate without friction without allowingvertical motion. The rod is rotating with angular velocity Ω with respect to the fixed inertial frame. Thearms are of length L. The frame Axyz rotates with the rod-masses system. Note that the angle φ isconstant.

Figure 3: Figure for Problem 4

a) Compute the angular momentum ~H of the two-mass system with respect to A.

b) Express the angular velocity ~ω of the rod-masses system as a vector using Axyz rotating unit vectors.

c) Express ~H as [I]{w} where [I] is a 3× 3 matrix and ~ω = {ω} = [wx wy wz]T .

d) Compute d ~H/dt. (Note that Ω is not assumed to be constant)

e) Find the torque about A and express it as a vector [τx τy τz]T .

g) Where could you place a single mass such that d ~H/dt would only have a torque component alignedwith ~ω?

f) Online survey question: What is the direction of the torque about A?

(a) K̂ (b) other but constant (c) it changes in time

Problem 5

A particle of mass m slides down a frictionless surface and collides with a uniform vertical rod of mass Mand length l, sticking to it. Following the collision, the rod pivots about the point O. G is the center ofmass of the rod.

Figure 4: Figure for Problem 5 (from Williams textbook)

a) In Williams textbook p283-284 you can find the expression of the inertia matrix for a slender rod aboutits center of mass. Use the concept of angular momentum about O to find θ̇ immediately after the collision.

b) Online survey question: Is the angular momentum ~H about O conserved during the collision?

(a) Conserved (b) not conserved (c) cannot tell

c) Online survey question: Is the linear momentum ~p conserved during the collision?

(a) Conserved (b) not conserved (c) cannot tell

Problem 6

The four-wheel drive truck sketched in the Figure has a mass M = 3000 kg and its center of mass is located2.5 m behind the front axle and 0.8 m above ground. The truck is carrying a crate of mass m = 800 kg onits flat bed. The static coefficient of friction between the truck’s bed and the crate is µs = 0.3.

Figure 5: Figure for Problem 6 (from Williams textbook)

a) Find the maximum forward acceleration of the truck that does not cause the crate to slip nor tip.

b) Online survey question: Is it necessary to know the mass moments of inertia for the box to solve thisproblem?

(a) yes (b) no

Problem 7

A pendulum consists of a thin rectangular plate (of thickness t) made of a material of density ρ with twoidentical circular holes (of radius R). The pivot is in O (fixed). Assume that all the dimensions shown aremuch greater than the uniform thickness t.

Figure 6: Figure for Problem 7 (from Williams textbook)

a) Find G the center of mass of the pendulum. Compute GIzz and OIzz. (Note: you can use the tablep283-284 in Williams)

b) Derive the equations of motion for the system. (Note: do not assume small motions)

Problem 8

Two uniform cylinders of respective mass m1 and m2 and radius R1 and R2 are welded together. Thiscomposite object rotates without friction about a fixed point O. An inextensible massless string is wrappedwithout slipping around the larger cylinder. The two ends of the string are connected to the ground via,respectively, a spring of constant k and a dashpot of constant c. The smaller cylinder is connected to ablock of mass m0 via an inextensible massless strap wrapped without slipping around the cylinder. Theblock is constrained to move only vertically.

Figure 7: Figure for Problem 8 (from Williams textbook)

a) Draw a free body diagram for the system.

b) Derive the equations of motion for the system.

c) Online survey question: How many independent coordinates are required to completely describe themotion of the system?

(a) 1 (b) 2 (c) 3 (d) 4

Problem 9

A wheel of weight 15 kg and a radius of gyration of kG = 0.6 m if placed on an inclined plane.

Figure 8: Figure for Problem 9

a) If the coefficients of static and kinetic friction between the wheel and the plane are µs = 0.2 and µk = 0.15respectively, determine the maximum angle θ of the inclined plane so that the wheel rolls without slipping.

b) Online survey question: If the tire begins with zero speed and travels down the hill, will it get to thebottom faster if µ = 0 and there is always slip, or if there is no slip?

(a) Faster with slip (b) Faster with no slip (c) Doesn’t matter