In the figure the coefficient of static friction between mass (MA) and the table is 0.40, whereas the coefficient of kinetic friction is 0.28 ?
part a) What minimum value of (MA) will keep the system from starting to move?
part b) What value of(MA) will keep the system moving at constant speed?
To solve part a and part b of this question, we need to understand the concepts of static friction and kinetic friction.
Static friction is the force that opposes the initiation of sliding motion between two surfaces in contact with each other and acts when the object is at rest. On the other hand, kinetic friction is the force that opposes the relative motion between two surfaces in contact with each other and acts when the object is in motion.
To find the minimum value of mass (MA) that will keep the system from starting to move (part a), we need to consider that the maximum static friction force is given by the equation:
Fstatic = μs * Fn
Where μs is the coefficient of static friction and Fn is the normal force. In this case, the normal force acting on mass (MA) is equal to its weight, which is MA * g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
So, the maximum static friction force is:
Fstatic = μs * (MA * g)
To prevent the system from starting to move, the maximum static friction force must be equal to the force acting on the system, which is the weight of both masses:
Fstatic = Fgravity
μs * (MA * g) = (MA + 2) * g
Simplifying the equation:
μs * MA = MA + 2
Rearranging the equation:
MA * (μs - 1) = 2
MA = 2 / (μs - 1)
Substituting the given value for the coefficient of static friction (μs = 0.40):
MA = 2 / (0.40 - 1)
MA = 2 / (-0.60)
MA = -3.33
However, mass cannot be negative, so the answer is invalid. Therefore, there is no minimum value of mass (MA) that will prevent the system from starting to move. The system will start moving no matter the value of (MA).
Now, let's move on to part b.
To find the value of mass (MA) that will keep the system moving at a constant speed, we need to consider that in this case, the friction force acting on the system is kinetic friction force:
Fkinetic = μk * Fn
Where μk is the coefficient of kinetic friction and Fn is the normal force.
Again, the normal force acting on mass (MA) is MA * g, so the kinetic friction force is:
Fkinetic = μk * (MA * g)
To keep the system moving at constant speed, the kinetic friction force must be equal to the force acting on the system (the weight of both masses):
Fkinetic = Fgravity
μk * (MA * g) = (MA + 2) * g
Simplifying the equation:
μk * MA = MA + 2
Rearranging the equation:
MA * (μk - 1) = 2
MA = 2 / (μk - 1)
Substituting the given value for the coefficient of kinetic friction (μk = 0.28):
MA = 2 / (0.28 - 1)
MA = 2 / (-0.72)
MA = -2.78
Again, mass cannot be negative, so the answer is invalid. There is no value of mass (MA) that will keep the system moving at a constant speed.