How Does Kinetic Energy Affect Needle Penetration in Medical Procedures?

[Zynoakib]

Homework Statement

In a needle biopsy, a narrow strip of tissue is extracted
from a patient using a hollow needle. Rather than
being pushed by hand, to ensure a clean cut the needle
can be fired into the patient’s body by a spring. Assume
that the needle has mass 5.60 g, the light spring has
force constant 375 N/m, and the spring is originally
compressed 8.10 cm to project the needle horizontally
without friction. After the needle leaves the spring,
the tip of the needle moves through 2.40 cm of skin
and soft tissue, which exerts on it a resistive force of
7.60 N. Next, the needle cuts 3.50 cm into an organ,
which exerts on it a backward force of 9.20 N. Find
(a) the maximum speed of the needle and (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
to 5.90 cm.

Homework Equations

The Attempt at a Solution

I am having trouble understanding (b)

My original attempt was :

Find the work done by friction:
(7.6)(0.024) + (9.2)(0.035) = 0.5044 J

and then I used KE = 1/2mv^2 to find the velocity because I thought in order for the needle to stop exactly at 5.90 cm the initial KE should be the same as the work done by friction. So, why am I wrong? Is it because if KE = work done by friction then there will be no energy for the needle to move?

Also, I want to ask if an object is already moving and then the KE is changed to a value that is the same as that of the work done by friction. Then, is the object moving at constant velocity?
Thank you!

 

[Ellispson]

Homework Statement

In a needle biopsy, a narrow strip of tissue is extracted
from a patient using a hollow needle. Rather than
being pushed by hand, to ensure a clean cut the needle
can be fired into the patient’s body by a spring. Assume
that the needle has mass 5.60 g, the light spring has
force constant 375 N/m, and the spring is originally
compressed 8.10 cm to project the needle horizontally
without friction. After the needle leaves the spring,
the tip of the needle moves through 2.40 cm of skin
and soft tissue, which exerts on it a resistive force of
7.60 N. Next, the needle cuts 3.50 cm into an organ,
which exerts on it a backward force of 9.20 N. Find
(a) the maximum speed of the needle and (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
to 5.90 cm.

Homework Equations

The Attempt at a Solution

I am having trouble understanding (b)

My original attempt was :

Find the work done by friction:
(7.6)(0.024) + (9.2)(0.035) = 0.5044 J

and then I used KE = 1/2mv^2 to find the velocity because I thought in order for the needle to stop exactly at 5.90 cm the initial KE should be the same as the work done by friction. So, why am I wrong? Is it because if KE = work done by friction then there will be no energy for the needle to move?

Also, I want to ask if an object is already moving and then the KE is changed to a value that is the same as that of the work done by friction. Then, is the object moving at constant velocity?
Thank you!

In your solution,I think you missed the change in potential energy of the spring.

 

[billy_joule]

the speed at which the flange on the back end of the needle
runs into a stop that is set to limit the penetration
to 5.90 cm.and then I used KE = 1/2mv^2 to find the velocity because I thought in order for the needle to stop exactly at 5.90 cm the initial KE should be the same as the work done by friction. So, why am I wrong?

You've misinterpreted the question. The flange hits the stop at some non-zero velocity, and all remaining energy is dispersed during this impact, it doesn't come to a rest due to friction. You need to find the KE just before impact to find the speed the question asks for.
That is:
E_spring - E_friction = E_{kinetic before impact}

The first term is given in the question, you've already calculated the second term so can find the third term and then find velocity before impact.

 

[Zynoakib]

You've misinterpreted the question. The flange hits the stop at some non-zero velocity, and all remaining energy is dispersed during this impact, it doesn't come to a rest due to friction. You need to find the KE just before impact to find the speed the question asks for.
That is:
E_spring - E_friction = E_{kinetic before impact}

The first term is given in the question, you've already calculated the second term so can find the third term and then find velocity before impact.

Thanks for the explanation, just two more question. If an object is already moving and then the KE is changed to a value that is the same as that of the work done by friction. Then, is the object moving at constant velocity?

Also, if the elastic PE of the spring is equal to the work done by friction in this problem. Then, what will happen to the needle?

 

[billy_joule]

Thanks for the explanation, just two more question. If an object is already moving and then the KE is changed to a value that is the same as that of the work done by friction. Then, is the object moving at constant velocity?

I can't quite parse that. Could you rephrase?
From Newtons first law, an object in motion stays in motion with the same velocity and in the same direction unless acted upon by an unbalanced force. Whatever happened in the past has no effect on this.
Generally, if friction is acting, (unpowered) objects slow down as an unbalanced force is present due to that friction.

Also, if the elastic PE of the spring is equal to the work done by friction in this problem. Then, what will happen to the needle?

The needle has stopped, all initial energy has been converted to heat via friction so no kinetic energy remains.

 

[Zynoakib]

The needle has stopped, all initial energy has been converted to heat via friction so no kinetic energy remains.

So, if there is no remaining KE in the needle, it stops and if there is remaining KE in the needle, the needle still stops at the same distance but those energy will be dispersed? Is this correct?

 

[billy_joule]

So, if there is no remaining KE in the needle, it stops and if there is remaining KE in the needle, the needle still stops at the same distance but those energy will be dispersed? Is this correct?

If something is moving it has KE. If something is not moving it has no KE. The needle can not both have KE and be motionless as the underlined section may imply.

In the case of your spring loaded needle there are two possible outcomes:
1)The work done by friction is beyond some critical value and the needle comes to stop before 5.90cm. All the KE was converted to heat via friction. No impact occurs.

2)This is the case the question is concerning. The work done by friction is less than that critical value and the needle still has KE at 5.90cm, the needle hits the flange and comes to a stop. The KE is dispersed in the impact, likely by an audible 'click', some imperceptible heating, maybe some microscopic deformation.

This is analogous to a car braking to avoid a collision with a brick wall. If the friction is high (dry road & good tyres) the car will disperse it's KE (and stop) before impact with the wall.
If friction is low (wet road, poor tyres) the cars will reach the wall with KE remaining. Upon impact, the KE will disperse by more violent means than hot brakes, eg: https://en.wikipedia.org/wiki/Crumple_zone

 

[Zynoakib]

If something is moving it has KE. If something is not moving it has no KE. The needle can not both have KE and be motionless as the underlined section may imply.

In the case of your spring loaded needle there are two possible outcomes:
1)The work done by friction is beyond some critical value and the needle comes to stop before 5.90cm. All the KE was converted to heat via friction. No impact occurs.

2)This is the case the question is concerning. The work done by friction is less than that critical value and the needle still has KE at 5.90cm, the needle hits the flange and comes to a stop. The KE is dispersed in the impact, likely by an audible 'click', some imperceptible heating, maybe some microscopic deformation.

This is analogous to a car braking to avoid a collision with a brick wall. If the friction is high (dry road & good tyres) the car will disperse it's KE (and stop) before impact with the wall.
If friction is low (wet road, poor tyres) the cars will reach the wall with KE remaining. Upon impact, the KE will disperse by more violent means than hot brakes, eg: https://en.wikipedia.org/wiki/Crumple_zone

Thanks for your explanations in great details, I was just wondering where the remaining KE go after the impact. Now I have no problems