Concept explainers
A 1.2-kg collar can slide along the rod shown. It is attached to an elastic cord anchored at F. The cord has an undeformed length of 300 mm and a spring constant of 70 N/m. Knowing that the collar is released from rest at A and neglecting friction, determine the speed of the collar (a) at B, (b) at E.
Fig. P13.64
(a)
Find the speed of the collar at B
Answer to Problem 13.64P
The speed of the collar at B
Explanation of Solution
Given information:
The mass of the collar (m) is
The un-deformed length
The spring constant (k) is
The length of AB
The length of AE
The length of DF
The acceleration due to gravity (g) is
Calculation:
Show the figure with the velocity as in Figure (1).
Calculate the elongated length of the elastic chord when the collar is at position A
Substitute
Calculate the elongated lengths of the elastic chord when the collar is at position B
Substitute
Calculate the elongation of the elastic cord when the collar is at position A
Substitute
Calculate the elongation of the elastic cord when collar is in position B
Substitute
Calculate the potential energy stored in the cord due to elongation
Substitute
Calculate the potential energy stored in the collar due to gravitation
Substitute
Substitute
Calculate the potential energy stored in the elastic cord when collar is at position A
Substitute
Calculate the potential energy stored in the cord due to elongation
Substitute
Calculate the potential energy stored in the collar due to gravitation
Substitute
Substitute
Calculate the potential energy stored in the elastic cord when collar is at position B
Substitute
Here, the collar is at rest at position A. Hence,
Calculate the kinetic energy of the collar at position B
Here, velocity of the collar at position B is
Substitute
Use the principle of conservation of energy to collar at position A and position B, to calculate the velocity of collar at position B. The principle of conservation of energy states that sum of the kinetic and potential energy of a particle remains constant.
Find the speed of the collar at B
Substitute 0 for
Therefore, the speed of the collar at B
(b)
Find the speed of the collar at E
Answer to Problem 13.64P
The speed of the collar at E
Explanation of Solution
Given information:
The mass of the collar (m) is
The un-deformed length
The spring constant (k) is
The length of AB
The length of AE
The length of DF
The acceleration due to gravity (g) is
Calculation:
Calculate the elongated length of the elastic chord when the collar is at position E
Substitute
Calculate the elongation of the elastic cord when collar is in position E
Substitute
Calculate the potential energy stored in the cord due to elongation
Substitute
Calculate the potential energy stored in the collar due to gravitation
Substitute
Calculate the potential energy stored in the elastic cord when collar is at position E
Substitute
Calculate the kinetic energy of the collar at position E
Here, velocity of the collar at position E is
Substitute
Apply the principle of conservation of energy to collar at position A and position E to calculate the velocity of the collar at position E.
Find the speed of the collar at E
Substitute 0 for
Therefore, the speed of the collar at E
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Chapter 13 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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