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?
Block on the table m(A) = m1,
block on the cord m2,
the coefficient of static friction is k1=0.4,
the coefficient of kinetic friction is k2 =0.28
(a)
Block A:
T = F(fr) = k1 •N = k(s) • m1 •g,
Block B: T = m2•g.
k1 • m1 •g= m2•g,
m1 = m2/k(s) = m2/0.4.
(b)
Block A:
T = F(fr) = k2 •N = k2 • m1 •g,
Block B:
T = m2•g.
k2• m1 •g= m2•g,
m1 = m2/k2 = m2/0.28.
To determine the minimum value of MA that will keep the system from starting to move, we need to consider the forces acting on the system.
Part a:
The force of static friction (Fs) can be calculated using the formula: Fs = μs * N
where μs is the coefficient of static friction and N is the normal force.
In this case, to prevent the system from starting to move, the static frictional force should be equal to the maximum force that could cause motion, which is the weight force (mg) of the system:
Fs ≥ mg
The normal force (N) is equal to the weight force (mg) since the system is on a table and is not accelerating vertically:
N = mg
So, substituting N = mg into the formula for static friction, we have:
Fs = μs * mg
Therefore, to prevent the system from starting to move:
μs * mg ≥ mg
μs ≥ 1
The minimum value of MA that will keep the system from starting to move is 1.
Part b:
To keep the system moving at a constant speed, the applied force (Fapplied) must be equal to the force of kinetic friction (Fk).
The force of kinetic friction (Fk) can be calculated using the formula: Fk = μk * N
where μk is the coefficient of kinetic friction and N is the normal force.
In this case, the normal force (N) is equal to the weight force (mg):
N = mg
So, substituting N = mg into the formula for kinetic friction, we have:
Fk = μk * mg
Therefore, to keep the system moving at a constant speed:
Fapplied = Fk
Fapplied = μk * mg
Since the system is not accelerating vertically, the gravitational force (mg) is balanced by the normal force (N = mg). Therefore, we can write the equation as:
Fapplied = μk * N
Substituting the values of μk and N, we have:
Fapplied = 0.28 * mg
Therefore, to keep the system moving at a constant speed, the value of MA should be such that:
MA * g = Fapplied
MA = (0.28 * mg) / g
MA = 0.28m
The value of MA that will keep the system moving at a constant speed is 0.28m.
To find the minimum value of MA that will keep the system from starting to move, we need to compare the force of static friction (FS) with the maximum possible force that can be exerted by MA.
The force of static friction can be calculated using the formula:
FS = μs * N
Where:
- μs is the coefficient of static friction (0.40 in this case)
- N is the normal force exerted on the object (equal to the weight of the object, which is the product of MA and the acceleration due to gravity, g)
The maximum possible force that can be exerted by MA is equal to the force of static friction when the object is on the verge of starting to move. Therefore, the equation becomes:
FS = MA * g
Setting these two equations equal to each other, we get:
MA * g = μs * N
Simplifying further, we get:
MA = (μs * N) / g
To find the minimum value of MA, we need to find the normal force exerted on the object. In this case, the object is resting on a table, so the normal force is equal to the weight of the object:
N = MA * g
Substituting this value of N back into the equation for MA, we get:
MA = (μs * (MA * g)) / g
Simplifying further, we find:
MA = μs * MA
Now, we can solve for MA by rearranging the equation:
MA - μs * MA = 0
Factoring out MA, we get:
MA * (1 - μs) = 0
Since we want to find the minimum value of MA, we need to find the value of MA when the expression (1 - μs) is equal to 0. Therefore, the minimum value of MA will be when (1 - μs) = 0, or when μs = 1.
So, the minimum value of MA that will keep the system from starting to move is MA = 1.
To find the value of MA that will keep the system moving at constant speed, we need to compare the force of kinetic friction (FK) with the force exerted by MA.
The force of kinetic friction can be calculated using the formula:
FK = μk * N
Where:
- μk is the coefficient of kinetic friction (0.28 in this case)
- N is the normal force exerted on the object (which is equal to the weight of the object, MA * g)
The force exerted by MA is equal to MA * g.
For the system to move at constant speed, the force exerted by MA must exactly balance the force of kinetic friction. Therefore, the equation becomes:
MA * g = FK
Setting these two equations equal to each other, we get:
MA * g = μk * N
Substituting the value of N as MA * g, we get:
MA * g = μk * (MA * g)
We can simplify this equation as:
MA * g = μk * MA * g
The g term can be canceled on both sides of the equation:
MA = μk * MA
Dividing both sides by MA, we find:
1 = μk
Therefore, the value of MA that will keep the system moving at constant speed is MA = 1.