A block of mass m1 is on top of a block of mass m2. Block 2 is connected by an ideal rope passing through a pulley to a block of unknown mass m3 as shown. The pulley is massless and frictionless. There is friction between block 1 and 2 and between the horizontal surface and block 2. Assume that the coefficient of kinetic friction between block 2 and the surface, μ, is equal to the coefficient of static friction between blocks 1 and
To find the unknown mass, m3, we need to analyze the forces acting on the system. Let's go step by step.
1. Identify the main forces: In this scenario, we have three key forces to consider:
a) The weight force (mg) acting downwards on each block, where m denotes mass and g is the acceleration due to gravity.
b) The normal force (N) exerted by the surface, perpendicular to it.
c) The friction force (f) acting in the direction opposite to the motion.
2. Determine the motion: Since block 1 is on top of block 2, the friction force between them will resist the movement of block 1. Therefore, the motion of the system is caused by the force between block 2 and the surface.
3. Calculate the net force: Newton's second law states that the net force acting on a body is equal to the product of its mass and acceleration. In this case, the net force can be determined by subtracting the frictional force from the weight force acting on block 2.
Net force on block 2 = weight force on block 2 - friction force on block 2
4. Express the forces in terms of masses and known variables:
a) The weight force on block 2 (mg2) can be expressed as m2g, where m2 is the mass of block 2.
b) The friction force (f) can be calculated by multiplying the coefficient of kinetic friction (μ) with the normal force (N). The normal force can be determined by taking into account the forces acting on block 1.
5. Apply Newton's second law:
Write the equation using the net force on block 2, and set it equal to the product of the mass of block 2 (m2) and the acceleration of the system (a).
m2g - μN = m2a
6. Consider the forces acting on block 1:
The friction force acting between block 1 and block 2 is equal in magnitude but opposite in direction to the friction force on block 2. Therefore, we can express it as μN.
7. Calculate the normal force on block 1:
The normal force exerted by block 1 is equal to the weight force on block 1, which is m1g.
8. Substitute the normal force into the equation:
m2g - μ(m1g) = m2a
9. Solve for the acceleration:
Rearrange the equation to solve for the acceleration (a).
a = (m2g - μ(m1g)) / m2
10. Determine the unknown mass, m3:
The mass m3 does not affect the acceleration of the system since it is hanging freely. So, to find m3, you can observe the acceleration already calculated and use Newton's second law with m3 to find the net force acting on it, which is equal to m3 times the acceleration.
m3g - T = m3a
Here, g represents the gravitational acceleration, and T is the tension in the rope.
11. Solve for m3:
Rearrange the equation to solve for m3.
m3 = (T + m3g) / a
Now, you have the steps to calculate the unknown mass, m3, in the system.