Concepts in Newtonian mechanics

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1 Concepts in ewtonian mechanics For each question, fill in the blank, circle true or false, circle one (or more) of the multiple choice answers, write the definition, or complete the drawing. The percentages of correct answers are from the pretest for 153 students in Stanford University s Advanced Dynamics classes in 2004, 2005, 2006, 2007, 2008, and Harvey Mudd students in % The product rule for differentiation that works when u and v are scalars, vectors, or matrices, is d(u v) = du v + u dv d(u v) = u dv + v du d(u v) = v du + u dv 97% Two properties (attributes) of a vector are and.??% The zero vector 0 has a magnitude of 0/1/2/ and no/one/all/any direction(s). All zero vectors are equal..??% Shown below are various vector operations (e.g., scalar multiplication, addition, dot-product, etc.). Circle those operations that are defined for a position vector a (with units of m) and a velocity vector b (with units of m sec ). -a 5 a a/5 a + b a b a b 78% Write the definition of the dot-product between a vector a and a vector b. Include a sketch with each symbol in your definition clearly labeled. a b = 41% Write the definition of the cross-product between a vector a and a vector b. Include a sketch with each symbol in your definition labeled and described. a b = where 89% The following questions apply to a vector a and a vector b: When a is parallel to b: a b =0 a b = 0 When a is perpendicular to b: a b =0 a b = 0 For arbitrary vectors a and b: a b = b a a b = b a 72% For arbitrary non-zero vectors a, b, c: a (b c) =(a b) c ever/sometimes/always A property of the scalar triple product is a b a = % Write a definition of v 2 in terms of v. v 2 =??% The cross-product of vectors a and b can be written in terms of a real scalar s as a b = s u where u is a unit vector perpendicular to both a and b in a direction dictated by the right-hand rule. A colleague tells you that the coefficient s of the unit vector u is inherently non-negative. Your colleague is correct/wrong. Copyright c by aul Mitiguy 269

2 41% Form the unit vector u having the same direction as c a x,wherea x is a unit vector and c is a real number. u = 95% Draw a right-handed orthogonal (mutually perpendicular) basis consisting of the unit vectors a x, a y, a z. 85% The vectors v 1 = x a x +2a y +3a z and v 2 =4a x +5a y +6a z are expressed in terms of orthogonal unit vectors a x, a y, a z. Find the value of x so v 1 and v 2 are perpendicular. x = 42% The column matrix is identical to the vector a x +2a y +3a z..??% r S/, the position vector of an object S from a point is to be determined. In general and without ambiguity, S could be a (circle all appropriate objects): Scalar Real number Complex number Center of a circle Vector oint Reference Frame Mass center of a set of particles Dyadic Set of oints Rigid Body Mass center of a rigid body Matrix article Flexible Body Set of flexible bodies rthogonal unit basis Set of articles Set of Rigid bodies System of particles and bodies 72% The following rotation matrix R relates two right-handed, orthogonal, unitary bases. Calculate its exact inverse by-hand (no calculator) in less than 30 seconds R = R -1 = ??% The following rotation table a R b relates right-handed, orthogonal, unit vectors a x, a y, a z and b x, b y, b z. Calculate the angle between a x and b z to four (or more) significant digits. a R b b x b y b z a x a y a z a x a z a y b z b y b x (a x, b z ) = 46% The following vectors are expressed in terms of the orthogonal unit vectors a x, a y, a z and t time. Circle the vectors that can be differentiated without consideration of a reference frame. 0 2 a x +4a y 2 a x + t a y a x 2 a x +4a y +6a z 2 a x + t a y +sin(t) a z 69% The definition of angular velocity of ω = θ k is a functional operational definition, i.e., in general, it is useful for calculating angular velocity and proving its properties (2D or 3D). Copyright c by aul Mitiguy 270

3 19% ω S, the angular velocity of an object S in a reference frame is to be determined. In general and without ambiguity, S could be a (circle all appropriate objects): Real number oint Reference Frame Mass center of a set of particles Complex number Set of oints Rigid Body Mass center of a rigid body Vector article Flexible Body Set of flexible bodies Matrix Set of articles Set of Rigid bodies System of particles and bodies 9% v S, the velocity of an object S in a reference frame is to be determined. In general, S should be a [circle all appropriate objects]: Scalar Vector Dyadic Matrix oint article Reference Frame Rigid Body 39% oints o and are fixed on rigid body B. Thevelocityv of in is defined as the time-derivative in of r (the position vector from o to ). Show that v canbewrittenintermsof ω B (the angular velocity of B in ) andr. Explain each step in your mathematical proof with a brief phrase. r B o Mathematical statement v = d r Reasoning Definition of velocity of in = ω B r 44% The following figures show a point moving in a plane. oint o is fixed in. The figure on the left shows moving in a clockwise direction with speed 12 on a circle of radius 4 that is centered at o. The figure on the right shows moving with a speed of 12 on a horizontal line that is 4 from o. Box the following true statements about angular velocity. s angular velocity in is nonzero. s angular velocity in is. s angular velocity in is 0. s angular velocity in does not exist. s angular velocity in is nonzero. s angular velocity in is. s angular velocity in is 0. s angular velocity in does not exist. 4 o 12 4 x 12 o 44% The following figures show a massive particle moving in a plane. oint o is fixed in. The figure on the left shows moving clockwise with speed 12 on a circle of radius 4 that is centered at o. The figure on the right shows moving with a speed of 12 on a horizontal line that is 4 from o. Box the following true statements about s angular momentum in. Copyright c by aul Mitiguy 271

4 s angular momentum about o is nonzero. s angular momentum about o is. s angular momentum about o is 0. s angular momentum about o does not exist. s angular momentum about o is nonzero. s angular momentum about o is. s angular momentum about o is 0. s angular momentum about o does not exist. 4 o 12 4 x 12 o 89% The figure to the right shows a point moving on a circle with a constant speed on a circle centered at o ( o is fixed in plane ). The following questions refer to v ( s velocity in ) and a ( s acceleration in ). The magnitude of a is constant a is constant in v is constant in o v 51% The figure to the right shows a thin circular disk C that remains in contact with a horizontal plane. ThepointofC that is in contact with at the instant this picture was taken is denoted C. For the questions that follow, remember that both rolling and sliding imply some kind of motion of C in. C C When C slides on, thevelocityofc in must be zero. When C rolls on, thevelocityofc in must be zero. When C slides on, the acceleration of C in can be zero. When C rolls on, the acceleration of C in can be zero. 3% α S, the angular acceleration of an object S in a reference frame is to be determined. S should be a [circle all appropriate objects]: Vector oint Reference Frame Mass center of a set of particles Dyadic Set of oints Rigid Body Mass center of a rigid body Matrix article Flexible Body Set of flexible bodies rthogonal unit basis Set of articles Set of Rigid bodies System of particles and bodies 33% K S, the kinetic energy of S in a reference frame is to be determined. In general, S should be a [circle all appropriate objects]: Vector oint Reference Frame Center of mass of a set of particles Dyadic Set of oints Rigid Body Center of mass of a rigid body Matrix article Flexible Body Set of flexible bodies rthogonal unit basis Set of articles Set of Rigid bodies System of particles and bodies 71% K S, kinetic energy of a system S in a reference frame always exists. 53% V S, potential energy of a system S in a reference frame always exists. Copyright c by aul Mitiguy 272

5 68% Write the definition for the moment of force F applied to point about point. Include a sketch with each part of your definition clearly labeled. M F / = 41% The resultant of a set of forces is a force. 78% In the SI (metric) system, the units of force are: 78% In the SI (metric) system, the units of impulse are: 58% The center of mass and center of gravity may be different points. 33% The center of mass and center of gravity of a rigid body always exists.??% ewton s law is violated if the resultant of forces on a massless object is non-zero??% ewton s law is violated if mass exists without the presence of force??% For F = ma to be valid, the m in mg must be exactly equal to the m in m a??% For Einstein s relativity to be valid, the m in mg must be exactly equal to the m in m a??% For modern string theory to be valid, the m in mg must be exactly equal to the m in m a 75% All torques are moments. 61% All moments are torques. 61% The moment of a couple about a point is equal to the moment of the couple about any other point 14% According to IST (ational Institute of Standards in Technology) 2007 data, the universal gravitational constant G is accurate to 2/4/8/16/ digits. 17% According to the Science magazine January 5, 2007 article (pg. 74) Atom Interferometer Measurement of the ewtonian Constant of Gravity, the IST universal gravitational constant G is accurate to 2/4/8/16/ digits. 51% M = Iα is useful for analyzing 3D rotational motions of a rigid body. 40% To thrust a satellite from low circular orbit about Earth to a higher circular orbit, an impulse is provided at two instants. The first impulse can be directed radially outward, tangent to the satellite s circular orbit, or directed at some angle θ 1 from the satellite s orbital tangent. The second impulse is applied at apogee (when the satellite is furthest from Earth) and is directed at an angle θ 2 from the orbital tangent. The first impulse puts the satellite into an elliptical orbit and the second changes the orbit from elliptical to circular. Using your engineering insights, provide values for θ 1 and θ 2 that minimize the amount of fuel required for this orbital transfer. Explain your reason for choosing these values. θ 1 = θ 2 = Reason: θ 2 θ 1 Copyright c by aul Mitiguy 273

6 11% Consider the following sets of forces. Circle the set(s) in which the following are all equal: ote: All forces have the same magnitude. Forces that are not horizontal or vertical are 30 from vertical. Moment of the set around point Moment of the set around point Moment of the set around point 57% Conceptual example of moments of inertia Each object below has a uniform density and an equal mass. After identifying the mass center of each figure with an, answer the following questions about I zz, the moment of inertia of each object about the line that passes through its mass center and is perpendicular to the plane of the paper. Flat disk Hollow flat disk Solid sphere Flat plate Hollow flat plate Solid block (a) Consider the first row of objects. The flat disk/hollow disk/solid sphere has the largest value of I zz,whereasthe flat disk/hollow disk/solid sphere has the smallest value of I zz. (b) Consider the second row of objects. The flat plate/hollow plate/solid block has the largest value of I zz,whereasthe and have equal values of I zz. (c) Consider all the objects in both rows. The has the largest value of I zz, whereas the has the smallest value of I zz. Copyright c by aul Mitiguy 274

7 ??% Concepts: What objects have a moment of inertia? Iuu S/, the moment of inertia of S about a point for the unit vector u is to be determined. (In other words, Iuu S/ is the moment of inertia of S about the line that passes through and is parallel to u.) In general, S should be a [circle all appropriate objects]: Vector oint Reference Frame Center of mass of a set of particles Dyadic Set of oints Rigid Body Center of mass of a rigid body Matrix article Flexible Body Set of flexible bodies rthogonal unit basis Set of articles Set of Rigid bodies System of particles and bodies 0% bjects A, B, C, D, ande are all flat planar objects with uniform density and the same mass. Denote S as one of A, B, C, D, ore. A B C D E n y n x n z 36% Consider Izz S/, the moment of inertia of S about the line passing through point and parallel to n z. List the objects in ascending order of Izz S/. If two or more objects have the same value of Izz S/, group them together. ote: The diameter of the circle and semi-circle, the wih of the square and rectangle, and the length of the thin rod are equal. Smallest Largest 28% List the objects in ascending order of Izz S/Scm, the moment of inertia of S about the line passing through S cm (the center of mass of S) and parallel to n z. If two or more objects have the same value of Izz S/Scm, group them together. Smallest D Largest 23% Consider Ixy S/, the product of inertia of S for the lines passing through point and parallel to n x and n y. For each object, determine if Ixy S/ is negative, zero, or positive and circle the corresponding symbol (, 0,+). A B C D E Copyright c by aul Mitiguy 275

8 16% roduct of inertia of a particle The position vector of a particle of mass m from a point is x n x +y n y, where n x, n y, n z are right-handed orthogonal unit vectors. I xy, the product of inertia of about for n x and n y can be calculated in terms of m, x, y, etc.,as I xy =. n y r / n x x (m) 8% Conceptual example of products of inertia The figure below shows six objects, each with a uniform density. For each object, consider I xy,the product of inertia of the object for lines that pass through point and are parallel to n x and n y. Below each object, mark whether the product of inertia is negative, zero, or positive. y n y nz n x Copyright c by aul Mitiguy 276

9 32% Conceptual example of translational motion The figure to the right shows a system consisting of a body B connected to a body A with a linear actuator. Initially, A and B are at rest (stationary) in deep empty space in a ewtonian (inertial) reference frame. A B It is possible for the linear actuator to move A s mass center in. It is possible for the linear actuator to move B s mass center in. It is possible for the linear actuator to move the system s mass center in. The previous three answers are the same/different if B is connected to A with a rotational motor (instead of a linear actuator). 32% Conceptual example of rotational motion The figure to the right shows a system consisting of a rigid body B connected to a rigid body A with a rotational motor (not shown). Initially, A and B are at rest (stationary) in deep empty space in a ewtonian (inertial) reference frame. A B It is possible for the rotational motor to rotate A in. It is possible for the rotational motor to rotate B in. Suppose the rotational motor is a revolute motor whose axis of rotation is initially parallel to a vector λ fixed in. Can the mass distribution of A cause this axis to point in another direction (i.e., so that at some later time the revolute axis is not parallel to λ)? Yes/o. Explain:??% Each rigid wheel below is pulled with a force F by a rope wrapped around its axle. The wheel starts from rest and is constrained to roll with a simple angular velocity. Determine which way the wheel rolls. The wheel rolls left/right The wheel rolls left/right ote: This problem was motivated by the dynamics of the old-fashioned penny farthing bicycle. Copyright c by aul Mitiguy 277

Flat disk Hollow disk Solid sphere. Flat plate Hollow plate Solid block. n y n O O O

Flat disk Hollow disk Solid sphere. Flat plate Hollow plate Solid block. n y n O O O NAME ME161 Midterm. Icertify that Iupheld the Stanford Honor code during this exam Tuesday October 29, 22 1:15-2:45 p.m. ffl Print your name and sign the honor code statement ffl You may use your course