Controlling a Rolling Ball On a Tilting Plane - STEM2

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Controlling a Rolling Ball On a Tilting Plane - STEM2

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Inthecaseofasphere

so the factor becomes

I = 2 5 Mr2 ,

M

M +(2/5)M = 5 7 .

16 Bringing the Ball to Rest in a Given Distance

by Tilting the Plane.

Given a position p and a velocity v, compute the pure incline plane tilt that

would give the ball a velocity −v after having traveled distance d. Thensince

the initial velocity was v, the ball will come to a zero velocity in distance d.

In practice the motion will not be exact, so the method can be iterated until

the desired result is obtained.

17 Given a Ball at Rest at Point P, Compute

the Tilt Motion to Bring It to Rest at

Point Q

Let d be the distance between P and Q. Give it a reasonable starting tilt in

the direction of Q, compute the time it takes to reach the halfway point. Use

the technique of the previous section to bring the ball to rest in distance d/2.

18 Making a Ball Follow a Specified Path

Given a curved path, compute sample points along the path. Do the tilting

to bring the ball to rest at each of the sample points.

19 Optimization

Not only must we move the ball in a specified path, we must do it in a short

amount of time. So more optimal methods of moving the ball will no doubt

be required.

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The kinetic energy of the ball is T = 1 2 Mv2 + 1 2 Iω2 The potential energy is where = 1 2 Mv2 + 1 2 I(v/r)2 = 1 2 (M + I/r2 )v 2 V = hgM, h = s sin(θ) is the height of the ball. The Lagrangian is L = T − V = 1 2 (M + I/r2 )v 2 − s sin(θ)gM. ∂L ∂v =(M + I/r2 )v d ∂L dt ∂v =(M + I/r2 ) dv dt ∂L ∂s = − sin(θ)gM. The equation of motion is So or d ∂L dt ∂v − ∂L ∂s =0 (M + I/r 2 ) dv dt dv dt +sin(θ)gM =0 M = −g sin(θ) M + I/r 2 So the effective acceleration has been reduced by the factor M M + I/r 2 14

Inthecaseofasphere so the factor becomes I = 2 5 Mr2 , M M +(2/5)M = 5 7 . 16 Bringing the Ball to Rest in a Given Distance by Tilting the Plane. Given a position p and a velocity v, compute the pure incline plane tilt that would give the ball a velocity −v after having traveled distance d. Thensince the initial velocity was v, the ball will come to a zero velocity in distance d. In practice the motion will not be exact, so the method can be iterated until the desired result is obtained. 17 Given a Ball at Rest at Point P, Compute the Tilt Motion to Bring It to Rest at Point Q Let d be the distance between P and Q. Give it a reasonable starting tilt in the direction of Q, compute the time it takes to reach the halfway point. Use the technique of the previous section to bring the ball to rest in distance d/2. 18 Making a Ball Follow a Specified Path Given a curved path, compute sample points along the path. Do the tilting to bring the ball to rest at each of the sample points. 19 Optimization Not only must we move the ball in a specified path, we must do it in a short amount of time. So more optimal methods of moving the ball will no doubt be required. 15

Page 1 and 2: Controlling a Rolling Ball On a Til
Page 3 and 4: = x 2 − 2xb + b 2 + a 2 − x 2 =
Page 5 and 6: 3 The Equation of a Plane Suppose a
Page 7 and 8: 6 The Mapping of Camera Coordinates
Page 9 and 10: where s 0 is the initial position.
Page 11 and 12: 12 A Mass Point in Circular Motion
Page 13: 14 The Simple Rotation of a Rigid B

Controlling a Rolling Ball On a Tilting Plane - STEM2

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