Concept explainers
The three-dimensional motion of a particle is defined by the cylindrical coordinates R = A/(t + 1), θ = Bt, and z = Ct/(t + 1). Determine the magnitudes of the velocity and acceleration when (a) t = 0, (b) t = ∞.
(a)
The magnitudes of the velocity
Answer to Problem 11.179P
The magnitudes of the velocity
Explanation of Solution
Given Information:
The three dimensional motion of a particle is defined by the cylindrical coordinates (R) is
Calculation:
The three dimensional motion of a particle is defined by the cylindrical coordinates (R):
The three dimensional motion of a particle is defined by the cylindrical coordinates
The three dimensional motion of a particle is defined by the cylindrical coordinates (z):
Differentiate the equation (1) with respective to time (t),
Differentiate the equation (4) with respective to time (t),
Differentiate the equation (2) with respective to time (t),
Differentiate the equation (5) with respective to time (t),
Differentiate the equation (3) with respective to time (t),
Differentiate the equation (6) with respective to time (t),
Calculate the value (R):
Substitute 0 for t in equation (1).
Calculate the value
Substitute 0 for t in equation (4).
Calculate the value
Substitute 0 for t in equation (4).
Calculate the value
Substitute 0 for t in equation (2).
Calculate the value (z):
Substitute 0 for t in equation (3).
Calculate the value
Substitute 0 for t in equation (7).
Calculate the value
Substitute 0 for t in equation (8).
Write the expression for radial component of velocity
Substitute A for
Write the expression for transverse component of velocity
Substitute A for R and
Write the expression for axial component of velocity
Substitute C for
Calculate the magnitude of the velocity (v) using the relation:
Substitute A for
Write the expression for radial component of acceleration
Substitute
Write the expression for transverse component of acceleration
Substitute A for R, B for
Write the expression for axial component of acceleration
Substitute
Calculate the magnitude of the acceleration (a) using the relation:
Substitute
Therefore, the magnitudes of the velocity (v) and acceleration (a) when time (t) is 0 are
(b)
The magnitudes of the velocity
Answer to Problem 11.179P
The magnitudes of the velocity
Explanation of Solution
Given Information:
The three dimensional motion of a particle is defined by the cylindrical coordinates (R) is
Calculation:
Calculate the value (R):
Substitute
Calculate the value
Substitute
Calculate the value
Substitute
Calculate the value
Substitute
Calculate the value (z):
Substitute
Calculate the value
Substitute
Calculate the value
Substitute
Write the expression for radial component of velocity
Substitute 0 for
Write the expression for transverse component of velocity
Substitute 0 for R and
Write the expression for axial component of velocity
Substitute 0 for
Calculate the magnitude of the velocity (v) using the relation:
Substitute 0 for
Write the expression for radial component of acceleration
Substitute 0 for
Write the expression for transverse component of acceleration
Substitute 0 for R, B for
Write the expression for axial component of acceleration
Substitute 0 for
Calculate the magnitude of the acceleration (a) using the relation:
Substitute 0 for
Therefore, the magnitudes of the velocity
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Chapter 11 Solutions
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