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Old Mar 21st 2009, 07:52 PM   #1
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Car on an inclined plane

I am completely stuck on this question and I have read through my text book and cannot find any relevant references to information I need, can anyone help?

A car is driving up a slope at angle 16 degrees to the horizontal, trying to accelerate as much as possible. The static and kinetic friction coefficients are 0.90 and 0.80 respectively.
Find the magnitude of the maximum possible acceleration (assuming a sufficiently powerful engine).
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Old Mar 21st 2009, 10:37 PM   #2
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This is an interesting problem. It seems to be a question of when the tires would start to slip. Why would they slip?. If you draw the acceleration due to the engine as a vector parallel to the surface of the incline, it has an upward component that is normal to the surface of the plane and tends to cancel the component of gravitational acceleration that is pushing down normal to the plane. So the force of friciton is decreased. Can you put this into equations?
(I don't know why the information about the static coefficient of friction was given.)
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Old Mar 21st 2009, 11:52 PM   #3
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Slipping car

Since the car is moving, the coeff of kinetic friction has to be considered.
i.e. 0.8. THe force it generates must overcome the force of friction
f = 0.8 x mg cos 16 + mg x sin 16 (which is the component of its weight along the slope. The acceleration it would experience is therefore

f/m = (0.8 x mg cos 16 + mg x sin 16 )/m = (0.8 x g cos 16 + g x sin 16 )

However , this is the minimum value .From the question there is no upper limit (theoretically) for f since the engine can be arbitrarily powerful .
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Old Mar 22nd 2009, 12:27 AM   #4
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Consider the car on a frictionless surface. Can car's engine move the car ? No, because force by the engine is internal one where as according to Newton's second law ( F=ma ) force have to be external in order to produce accln. So we have to look for an external force. The only external force on the car appears to be the friction directed forward. So in this case friction causes the motion and not opposes it. Now the next question, it is static or kinetic friction. If you watch the tyre of a moving car you will see that it is rolling and not slipping on the surface. There is no relative motion between the earth surface and the tyre at the point of contact. So this is static friction that causes the motion of the car.
Now the maximum frictional force can be equal to " coefficient of static friction times the normal reaction" which is in your case is
k*mg*cos16
put this equal to ma and you get the maximum accln
a = k*g*cos16 ( k being coefficient of static friction)
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Old Mar 22nd 2009, 02:25 AM   #5
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If a car with an engine cannot move on a frictionless surface because there is no friction, then a car without an engine on a rough surface should move by itself because friction is present and we would have no fuel problems whatsoever! I dont think that logic holds because the force for motion has to be generated by the engine, gets coverted to torque which is doing work against the internal friction present in the axle etc.
If it is on a frictionless surface,the wheels slip that is all. Friction is a necessary but not sufficient condition for motion in this particular case.
It is said that tiny bonds are formed between the surfaces and that is why friction arises. When there is motion these are being formed and broken continuously, so the coeff of kinetic friction is less.
If there is an upper limit to the accln then a car should not be able to increase speed when going up a slope which is not the case in practice.
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Old Mar 22nd 2009, 06:49 AM   #6
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I agree friction is not the sufficient condition. Actually car's tyre exert a friction force backward and in reaction earth exert the friction force forward.
No matter how powerful is the engine car can not acclerate beyond this limit. Engineers try to increase the coefficient of friction by using special design and matterial in order to increase the accln.
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Old Mar 22nd 2009, 08:05 AM   #7
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Is this true only on a slope or also on level roads?
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Old Mar 22nd 2009, 09:01 AM   #8
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On the level road this limit is 'coefficient of static friction times accln due to gravity'
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