When the three blocks in the figure are released from rest, they accelerate with a magnitude of 0.900 m/s2. Block 1 has mass M, block 2 has 2M, and block 3 has 2M. What is the coefficient of kinetic friction between block 2 and the table?
What figure?
To find the coefficient of kinetic friction between block 2 and the table, we need to consider the forces acting on block 2.
Let's start by analyzing the forces on block 2. The force of gravity (weight) on block 2 is given by:
Fg = m * g
Where m is the mass of block 2 and g is the acceleration due to gravity.
Next, we need to find the net force acting on block 2. Considering the direction of acceleration, the force of friction acting on block 2 opposes its motion. So, the equation for the net force on block 2 is:
F_net = Fg - F_friction
Where F_friction is the force of kinetic friction.
Using Newton's second law, which states that the net force on an object is equal to the mass of the object multiplied by its acceleration, we can write:
F_net = m * a
Combining these equations, we have:
Fg - F_friction = m * a
Now, let's relate the force of friction to the normal force and the coefficient of kinetic friction. The force of friction can be expressed as:
F_friction = μ * N
Where μ is the coefficient of kinetic friction and N is the normal force (the force exerted by the table on block 2).
The normal force is equal to the weight (mg) of the blocks above it:
N = m * g + 2M * g
Now, let's substitute the values and simplify the equation:
(m * g + 2M * g) - (μ * (m * g + 2M * g)) = m * a
Simplifying further:
m * g + 2M * g - μ * (m * g + 2M * g) = m * a
Now, group the terms:
(m * g + 2M * g) * (1 - μ) = m * a
Divide both sides by (m * g + 2M * g):
1 - μ = (m * a) / (m * g + 2M * g)
Next, simplify the right side:
(m * a) / (m * g + 2M * g) = a / (g + 2M * g/m)
Now, let's substitute the given acceleration value of 0.900 m/s^2:
1 - μ = 0.900 / (g + 2M * g/m)
To find the coefficient of kinetic friction (μ), we need the value of the acceleration due to gravity (g). Assuming it is approximately 9.8 m/s^2, we can calculate μ.
Please note that the value of g might change depending on the units used, and you need to convert the units accordingly.
To find the coefficient of kinetic friction between block 2 and the table, we need to consider the forces acting on block 2.
First, let's analyze the forces on block 2:
1. Weight (mg): This is the force exerted by gravity on block 2, where m is the mass of block 2, and g is the acceleration due to gravity.
2. Normal force (N): This is the perpendicular force exerted by the table on block 2.
3. Friction force (f): This is the force due to kinetic friction acting on block 2, opposing its motion.
4. Tension force (T): This is the force acting on block 2, pulling it to the right, transmitted through the string from block 1.
Since the blocks are accelerating, we know that the net force acting on block 2 is non-zero. The net force is given by the equation:
Net force = ma
Where m is the mass of block 2 and a is its acceleration.
The net force acting on block 2 can be expressed as the sum of the forces mentioned above:
Net force = T - f - mg = ma
Rearranging the equation, we get:
f = T - mg - ma
Now, considering that the mass of block 2 is 2M and its acceleration is 0.900 m/s^2, we can substitute these values into the equation:
f = T - (2M)(g) - (2M)(a)
Since the problem states that the blocks are released from rest, the tension force (T) pulling block 2 is equal to the force of static friction (fs) between block 2 and block 1.
So, we have:
f = fs - (2M)(g) - (2M)(a)
We know that the force of static friction (fs) is given by:
fs = μs * N
Where μs is the coefficient of static friction between block 2 and block 1, and N is the normal force.
Since block 2 is not moving vertically, the normal force N is equal to the weight of block 2:
N = (2M)(g)
Substituting this into the equation for fs, we get:
fs = μs * (2M)(g)
Now, we can substitute this expression for fs into the equation for f:
f = (μs * (2M)(g)) - (2M)(g) - (2M)(a)
Simplifying further:
f = (μs * (2M)(g)) - (2M)(g) - (2M)(a)
To find the coefficient of kinetic friction (μk), we need to divide the friction force (f) by the normal force (N).
μk = f / N
Substituting the expression for f and N:
μk = [(μs * (2M)(g) - (2M)(g) - (2M)(a))] / [(2M)(g)]
Finally, we can cancel out the common factors of (2M)(g):
μk = [μs - 1 - a/g]
So, the coefficient of kinetic friction between block 2 and the table is given by μk = [μs - 1 - a/g].