
Concept explainers
(a)
The maximum speed of the needle.
(a)

Answer to Problem 78AP
The maximum speed of the needle is
Explanation of Solution
Consider the needle-spring-body as a system. Since there are resisting forces when the needle moves through the skin. Hence, the maximum speed of the needle when projected horizontally using a spring will be at the point on the patient body.
Consider, the spring-needle as an isolated system.
Since the total energy of the system is conserved from the point the spring is maximumly compressed to the point the needle is on the point on the body of the patient. Hence, the total kinetic and potential energy will be conserved.
Write the equation for conservation of energy
Here,
Write the expression for the change in spring potential energy
Here,
Write the equation for spring potential energy
Here,
Since the spring is compressed initially and when the needle touches the skin spring has no potential energy.
Substitute
Simplify the above equation.
Write the expression for the change in kinetic energy of the spring-needle system
Here,
Since the needle is the one having the kinetic energy change when the needle moves from spring to patient body.
Write the equation for the kinetic energy of the needle
Here,
Substitute
Simplify the above equation.
Substitute
Rearrange the above equation.
Simplify the above equation.
Conclusion:
Substitute
Thus, the maximum speed of the needle is
(b)
The speed at which the flange on the back end of the needle runs into a stop that is set to limit the penetration.
(b)

Answer to Problem 78AP
The speed at which the flange on the back end of the needle runs into a stop that is set to limit the penetration is
Explanation of Solution
Consider the needle-spring-body system as a system.
Since there are resisting forces when the needle moves through the skin.
Hence, the total energy of the needle while penetrating the skin is transformed into work done by the resisting forces.
Consider the needle-body as an isolated system.
Since the work done on the system is by internal resisting forces which are non-conservative.
Write the equation for conservation for energy
Here,
Since the needle moves horizontally, hence there is no change in potential energy.
Write the expression for work done
Here,
Write the equation for work done by resisting force
Here,
Write the expression for work done using the above equation
Here,
Since the needle is the one having the kinetic energy change when the needle moves from spring to patient body.
Substitute
Simplify the above equation.
Since the spring is compressed initially and when the needle touches the skin spring has no potential energy.
Substitute
Simplify the above equation.
Substitute
Rearrange the above equation.
Simplify the above equation.
Conclusion:
Substitute
Substitute
Thus, the speed at which the flange on the back end of the needle runs into a stop that is set to limit the penetration is
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