Consider the system of blocks in the figure below, with m2 = 5.0 kg and è = 33°. If the coefficient of static friction between block #1 and the inclined plane is ìS = 0.24, what is the largest mass m1 for which the blocks will remain at rest?

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To find the largest mass m1 for which the blocks will remain at rest, we need to consider the forces acting on the system.

First, let's analyze the forces acting on block m2. There are three forces acting on m2: its weight (mg2) acting vertically downward, the normal force (N) exerted by m1, and the friction force (f) between m2 and the inclined plane.

The weight of m2 is given by mg2, where g is the acceleration due to gravity (approximately 9.8 m/s^2) and m2 is the mass of the block (5.0 kg).

The normal force exerted by m1 can be found using trigonometry. Since the incline is at an angle of 33 degrees, the normal force N can be calculated as N = mg1 * cos(33°), where g1 is the acceleration due to gravity acting on m1 (also approximately 9.8 m/s^2).

The friction force f can be calculated using the equation f = ìS * N, where ìS is the coefficient of static friction between m1 and the inclined plane, and N is the normal force.

Now let's analyze the forces acting on block m1. There are two forces acting on m1: its weight (mg1) and the force of static friction (f) between m1 and the inclined plane.

Since the blocks are at rest, the net force on both blocks must be zero. Therefore, the force of static friction f acting on m2 must be equal to the force of static friction acting on m1.

Finally, to find the largest mass m1 for which the blocks will remain at rest, we can set up an equation using the forces of static friction on both blocks:

ìS * N = ìS * mg1 * cos(33°)

Simplifying the equation, we have:

mg1 * cos(33°) = mg2

Now we can solve for m1:

m1 = m2 / cos(33°)

Substituting the given values, we have:

m1 = 5.0 kg / cos(33°)

Using a calculator, we can find that cos(33°) is approximately 0.8387.

m1 = 5.0 kg / 0.8387 ≈ 5.96 kg

Therefore, the largest mass m1 for which the blocks will remain at rest is approximately 5.96 kg.