Rat Running on Ice: Solutions & Tips

[Vibhor]

Homework Statement

Homework Equations

The Attempt at a Solution

There is no tangential acceleration , only normal acceleration ,given by ν²/R .But I do not understand what is the radius of the curve since rat is moving along straight line before and after the turn (I think that is the case ) . The maximum static frictional force on rat is μmg .

How should I proceed with the problem .

Thanks

 

[andrewkirk]

Given the coefficient of friction you can express the maximum force the rat can exert to change direction, as a multiple of its mass. If it exerts that force constantly to its left until it has turned 90 degrees to the right, the force has generated a centripetal acceleration throughout that right turn. Find the radius at which that is the required centripetal acceleration to move in a circle at speed v. From there you should be able to work out the time taken.

 

[Vibhor]

Given the coefficient of friction you can express the maximum force the rat can exert to change direction, as a multiple of its mass.

Isn't the maximum force = μmg ?

 

[andrewkirk]

Isn't the maximum force = μmg ?

Yes, and that's $(\mu g)\times m$, which is a multiple of the mass.

 

[Vibhor]

the force has generated a centripetal acceleration throughout that right turn. Find the radius at which that is the required centripetal acceleration to move in a circle at speed v.

How to find the circle and hence the radius ? Rat hasn't moved on a curve ,it made a sharp 90° turn while moving in a straight line ( I suppose) .

 

[andrewkirk]

Rat hasn't moved on a curve ,it made a sharp 90° turn while moving in a straight line ( I suppose) .

That's not what the question means. It says the rat 'suddenly decides to turn', not that the rat 'suddenly turns'. The sudden-ness means there is a discontinuity in the second derivative of position (acceleration), not the first (velocity). So the rat is traveling in a straight line, then suddenly starts curving (say) right in the tightest circle it can manage without slipping, until it has turned 90 degrees, at which point it (suddenly) stops curving and continues on in a (new) straight line. The circle they're asking about is the quarter-circle path the rat follows while executing that change of direction.

 

[Vibhor]

That's not what the question means. It says the rat 'suddenly decides to turn', not that the rat 'suddenly turns'. The sudden-ness means there is a discontinuity in the second derivative of position (acceleration), not the first (velocity). So the rat is traveling in a straight line, then suddenly starts curving (say) right in the tightest circle it can manage without slipping, until it has turned 90 degrees, at which point it (suddenly) stops curving and continues on in a (new) straight line.

Ok .

Now , how should I find the radius of the curve ?

 

[andrewkirk]

Go to post 2.

 

[Vibhor]

Ok

 

[Vibhor]

How should I find the least distance covered by mice while making the turn such that time taken is minimum ?

 

[SammyS]

How should I find the least distance covered by mice while making the turn such that time taken is minimum ?

What have you found so far?

Did you find the radius?

 

[Vibhor]

Did you find the radius?

If I consider the curve to be a quarter circle ,then that does solve the problem . But this is what I do not understand , why the curve traced by rat should be a part of a circle ?

 

[ehild]

If I consider the curve to be a quarter circle ,then that does solve the problem . But this is what I do not understand , why the curve traced by rat should be a part of a circle ?

No sudden change of direction can happen. And any tiny part of a curve can be replaced by a circle. And friction allows a certain radius or greater. The speed is given, so the turn takes the shortest time if the radius is the shortest. The shortest radius along the whole path... it is a quarter circle, is not it?

 

[Vibhor]

The speed is given, so the turn takes the shortest time if the radius is the shortest. The shortest radius along the whole path... it is a quarter circle, is not it?

How would circular path ensure minimum distance ? Why can't it be any other curve ?

 

[andrewkirk]

THey want to change direction in minimum time. So they push sideways as hard as they can without slipping. THe limit to how hard they can push is fixed by their weight and the friction coefficient. So they push sideways at a constant rate of that fixed limit until their direction has changed by 90 degrees. A constant acceleration perpendicular to the direction of motion causes a body to move along an arc of a circle.

By the way, the rat may be able to change direction in a shorter distance and/or time by pushing against the direction of motion until it comes to a stop, then turning 90 degrees on the spot and setting off again. But this is forbidden by the requirement in the OP question that the speed remain constant. That in turn requires that any pushing must be perpendicular to the direction of motion.

 

[SammyS]

If I consider the curve to be a quarter circle ,then that does solve the problem . But this is what I do not understand , why the curve traced by rat should be a part of a circle ?

You yourself noted that the tangential acceleration is zero (the result of constant speed).

If you want the change in direction to occur in min. time, then it follows that the radial (normal, in this case) acceleration is a maximum, which is a constant value. Thus you have circular motion - for a portion of a circle.

Right?

(Is ehild awake and doing PF so early in the morning? Time for me to go to sleep!)

 

[ehild]

How would circular path ensure minimum distance ? Why can't it be any other curve ?

Read my previous post again. Any tiny part of a curve can be replaced by an arc of circle. For shortest time, you need the shortest arc. The minimum radius of that short arc of circle is determined by the speed and the coefficient of the friction. The next short arc has to be of the same radius. The next one too. A curve with parts all of the same radius is a circle.

 

[ehild]

(Is ehild awake and doing PF so early in the morning? Time for me to go to sleep!)

We must live about a quarter arc away on the Earth.
I get up at 6 usually, have my coffee and read PF.

 

[haruspex]

Read my previous post again. Any tiny part of a curve can be replaced by an arc of circle. For shortest time, you need the shortest arc. The minimum radius of that short arc of circle is determined by the speed and the coefficient of the friction. The next short arc has to be of the same radius. The next one too. A curve with parts all of the same radius is a circle.

As Andrew noted, the minimum time requirement does not of itself lead to a circular arc. First, you have to deduce that the acceleration must at all times be normal to the direction of travel in order to comply with the constant speed condition. Then, minimum time implies maximum normal acceleration, which implies constant magnitude.
Indeed, if the constant speed condition is removed then the quickest is to come to a stop as soon as possible.
The condition could be relaxed to "moving at the original speed but at 90 degrees to the original direction as quickly as possible". That would still lead to a circular arc, I believe, but it might be harder to prove that.

 

[Vibhor]

THey want to change direction in minimum time. So they push sideways as hard as they can without slipping. THe limit to how hard they can push is fixed by their weight and the friction coefficient. So they push sideways at a constant rate of that fixed limit until their direction has changed by 90 degrees. A constant acceleration perpendicular to the direction of motion causes a body to move along an arc of a circle.

By the way, the rat may be able to change direction in a shorter distance and/or time by pushing against the direction of motion until it comes to a stop, then turning 90 degrees on the spot and setting off again. But this is forbidden by the requirement in the OP question that the speed remain constant. That in turn requires that any pushing must be perpendicular to the direction of motion.

Excellent explanation andrew . Very nice !