Concept explainers
The end point B of a boom is originally 5 m from fixed point A when the driver starts to retract the boom with a constant radial acceleration of r = −1.0 m/s2 and lower it with a constant angular acceleration θ = −0.5 rad/ s2. At t = 2 s, determine (a) the velocity of point B,(b) the acceleration of point B, (c) the radius of curvature of the path.
Fig. P11.192
(a)
The velocity
Answer to Problem 11.192RP
The velocity
Explanation of Solution
Given information:
The distance
The boom with a constant radial velocity
The boom with a constant radial acceleration
The boom with a constant angular acceleration
The time (t) is 2 sec.
Calculation:
Write the expression for rectangular position coordinate (r) of point B using equation of motion:
Here,
The radial initial velocity of point B is 0.
Calculate the radial coordinate (r) of point B:
Substitute 5m for
Calculate the radial velocity
Substitute 0 for
Calculate the angular coordinate
Here,
Angular coordinate of initial velocity of point B is 0. Thus,
Substitute
Calculate the
Substitute zero for
Calculate velocity
Here,
Rewrite Equation (2) in terms of r,
Substitute
Calculate the magnitude of
Substitute
Calculate unit vector
Substitute
Calculate angle
Substitute
Calculate the angle
Substitute
Therefore, the velocity
(b)
The acceleration
Answer to Problem 11.192RP
The acceleration
Explanation of Solution
Given information:
The distance
The boom with a constant radial acceleration
The boom with a constant radial acceleration
The boom with a constant angular acceleration
The time (t) is 2 sec.
Calculation:
Show the values of
Write acceleration
Here,
Rewrite the above equation in term of r,
Substitute
Here,
Calculate the magnitude of
Substitute
Calculate the angle
Substitute
Calculate the angle
Substitute
Therefore, the acceleration
(c)
The radius of curvature
Answer to Problem 11.192RP
The radius of curvature
Explanation of Solution
Given information:
The distance
The boom with a constant radial acceleration
The boom with a constant radial acceleration
The boom with a constant angular acceleration
The time (t) is 2 sec.
Calculation:
Calculate the tangential component of acceleration
Substitute
Write normal component
Substitute
Substitute
Calculate the normal acceleration
Substitute
Calculate the radius of curvature
Rewrite Equation for radius of curvature.
Substitute
Therefore, the radius of curvature
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Chapter 11 Solutions
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