Please show me how to get these 2 answers In an inertial ENU-coordinate system,

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Please show me how to get these 2 answers

In an inertial ENU-coordinate system, a train accelerates in the East-direction with magnitude a. A passenger travels with his back towards the driving direction and holds a spinning wheel sideways using a rod going through the axle, such that the axis about which the wheel rotates points in the Up- direction. Around one of the spokes (Dutch: "spaken") of the wheel, a coloured marble is attached and starts moving at timet 0s. . The marble is moving outward from the center towards the rim (Dutch: "buitenrand") of the wheel with a relatice speed defined by vbt parameter b is a constant acceleration of b 0.1 m/s2 . the train accelerates with a = 2 mis2 the wheel spins with a constant angular velocity of magnitude wrad/s, initially, the marble is located at a distance r from the center of the wheel, pointing in the inertial North direction, with r = 10 cm. Determine the magnitude of the acceleration of the marble, relative to the inertial coordinate system at Incorrect Solution: Number Units Your Answer Correct A r: 2.1879 m/s*2 ± 2.0% Consider a thin rolling wheel with a body-fixed coordinate system zyz, associated with frame B and origin O. The z-axis passes through the centred axle and the z- and y-axes are fixed to the spokes. Let the wheel roll over the ground without slipping. Movements are considered relative to the inertial coordinate system XYZ, associated with frame N and origin O, where the XZ-plane is the ground. The wheel can rotate around all axes. The wheel is rolling in a perfect circular motion about the positive Y axis, completing one cycle with radius R, from O to the contact point of the wheel with the ground Consider the wheel in the following orientation Its z-axis is parallel to the ground, pointing into the direction of where the axle moves to The wheel is tilted about its z-axis, such that its z-axis points towards O The unit vector in the z-direction has no X-component. Give the component of angular velocity vector of the wheel, as expressed in the N frame if: the wheel's radius isr 1 m, . the circle around the Y-axis has radius R = 5 m, one revolution about the Y-axis is done in 20 seconds Incorrect Solution: Number Units Your Answer Correct Answer( .4495pi rad/s ± 2.0%

Explanation / Answer

given

in interial ENU coordiante system

train accelerates in east direction with magnitude a

hence at = ai

speed of passenger in thebackward direction = v

hence speed of passenger in inertial frame = -(v - u)i

where u is peed of the train in +i direction at that time

marble starts moving at t = 0

a. marble moves outwards from the center towards the rim with relative refernce speed v = bt

b = 0.1 m/s/s

a = 2 m/s/s

w = pi*rad/s

ri = r (pointing in inertial north direction), r = 10 cm

hence

acceleration of marble = a i + b j - (w^2*r)j

a = 2i + (0.1 - pi^2*0.1)j = 2i -0.8869604 j

|a| = sqrt(4 + -0.886^2) = 2.187 rad/s/s

b. thin rolling wheel

body fixed coordiante system xyz, frame B, origin O'

inertial coordinate system XYZ

wheel is rolling about Y axis, perfectly circular motion, radius R

x axis is paralle to the graound, pointing in the direction where wheel moves

z axis points towards O

unit vector in z direction has no X component

r = 1 m

R = 5 m

T = 20 s

hence

w = 2*pi/T = 0.31415926 rad/s

and

w' = ?

now

2*pi*R = n*2*pi*r

n = 5

hence

w' = 5*w = 1.5707963 rad/s

angular velocity of the wheel in xyz frame is

W = w'*sqrt(R^2 - r^2)/R [sin(wt)i + cos(wt)k] - w*r/R j + w j

Wz = w'*sqrt(R^2 - r^2)/R cos(wt)

at t= 0

wZ = 1.50665488 rad/s

Wz =

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