Learning Goal: To be able to find an object\'s angular velocity, linear velocity

Question

Learning Goal:

To be able to find an object's angular velocity, linear velocity, and acceleration in a fixed reference frame.

?=??

?=?1+?2

?=??

v=?×r

a=?×r+?×(?×r)

Part A

Part complete

?=?atan(1140m?250m/s t515m) radians

?=5.00×10?2 t radians

Express your answers numerically in radians per second to four significant figures separated by commas.

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5.000×10?2,0.0000,0.4854

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Correct

At any other value of ? besides zero, the camera's angular velocity would have a component about the y axis. Thus, when ? is zero, the airplane is over the y axis and the angle ? is a measure solely of the angle that the airplane rotates about the x axis.

Part B

Part complete

At t = 4.56 s , the airplane is r = 528 m away from the spy. (Figure 2) What are vx, vy, and vz, the i, j, and k components of the airplane's velocity vector v, when ?=0.000?

Express your answers numerically in meters per second to three significant figures separated by commas.

View Available Hint(s)

-250,-5.97,25.7

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Correct

Part C

At t = 4.56 s and ?=0.000, the airplane's angular acceleration is (0.000)i+(?2.43×10?2)j+(0.000)k radians/s2. (Figure 3) What are ax, ay, and az, the i, j, and k components of a, the airplane's acceleration? Recall that the airplane's position vector at P is r=0.000i+514j+119k m.

Express your answers numerically in meters per squared second to three significant figures separated by commas.

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3 of 3

Learning Goal:

To be able to find an object's angular velocity, linear velocity, and acceleration in a fixed reference frame.

When a rigid body undergoes an angular rotation, d?, about a fixed point, the body's angular velocity is the derivative of ? with respect to time:

?=??

where ? is the angular velocity vector and ?? is the time derivative of ?. If the body undergoes rotations in two directions, the body's angular velocity is the sum of the angular velocity vectors in each direction:

?=?1+?2

The body's angular acceleration is equal to the time derivative of the angular velocity:

?=??

where ? is the angular acceleration vector and ?? is the time derivative of ?. The velocity of any point P on the body rotating around the fixed point is given by

v=?×r

where v is the velocity vector of the point and r is the position vector of P measured from the fixed point about which the body rotates. If the angular velocity, ?, and angular acceleration, ?, are known at any instant, then the acceleration of point P is given by the time derivative of the velocity:

a=?×r+?×(?×r)

Part A

Part complete

A spy, who is surveilling an enemy's airfield, is photographing airplanes taking off. (Figure 1) One airplane takes off, banks hard, and circles the spy's location at a constant radius. As the spy tracks the airplane, the camera's telephoto lens rotates about the x, y, and z axes according to the equations

?=?atan(1140m?250m/s t515m) radians

and

?=5.00×10?2 t radians

, where t is time measured in seconds from takeoff. What are ?x, ?y, and ?z, the i, j, and k components of ?, the camera's angular velocity at t = 4.56 s , when ?=0.000?

Express your answers numerically in radians per second to four significant figures separated by commas.

View Available Hint(s)

?x, ?y, ?z =

5.000×10?2,0.0000,0.4854

radians/s

SubmitPrevious Answers

Correct

At any other value of ? besides zero, the camera's angular velocity would have a component about the y axis. Thus, when ? is zero, the airplane is over the y axis and the angle ? is a measure solely of the angle that the airplane rotates about the x axis.

Part B

Part complete

At t = 4.56 s , the airplane is r = 528 m away from the spy. (Figure 2) What are vx, vy, and vz, the i, j, and k components of the airplane's velocity vector v, when ?=0.000?

Express your answers numerically in meters per second to three significant figures separated by commas.

View Available Hint(s)

vx, vy, vz =

-250,-5.97,25.7

m/s

SubmitPrevious Answers

Correct

Part C

At t = 4.56 s and ?=0.000, the airplane's angular acceleration is (0.000)i+(?2.43×10?2)j+(0.000)k radians/s2. (Figure 3) What are ax, ay, and az, the i, j, and k components of a, the airplane's acceleration? Recall that the airplane's position vector at P is r=0.000i+514j+119k m.

Express your answers numerically in meters per squared second to three significant figures separated by commas.

View Available Hint(s)

ax, ay, az = m/s2

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Figure

3 of 3

u01 ?3

Explanation / Answer

Total acceleration = tangential acceleration + radial acceleration

Tang. ACC. = Ang. ACC. × r

(Angular acceleration is not visible in question clearly)

But Tang. ACC. =( 0.000i + 514j + 119k)× (ang. ACC.)

You will get the value for ax ay and az from this equation

Since ang. Velocity = 0 (given in question) so normal acceleration = 0

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