A block with mass M = 5.00 kg rests on a frictionless table and is attached by a

horizontal spring (k = 130 N/m) to a wall. A second block, of mass m = 1.25 kg rests on top of M.
The blocks are displaced by 10 cm then released.
If the coefficient of static friction between the two blocks
is 0.3, does block m slip off block M? (Show your work)
YES / NO

F = - k x

= -130 (.1) = -13 N

if they do not slip m = 6.25 kg

a = 13/6.25 = 2.08 m/s^2

force required to accelerate small block = 1.25 *2.08 = 2.6 N

do we have that much?
weight = m g = 1.25 * 9.81
so max friction = .3*1.25*9.81 = 3.68 N
so
no slip :)

Thank you so much!

You are welcome :)

Just to clarify, when you conclude that the block m doesn't slip, which forces are you comparing? If the friction was smaller than the force required to accelerate the small block, would it then slip?

Yes, if the friction was less than the force required to accelerate the upper block, it would accelerate more slowly than the bottom one. In other words slip :)

To determine whether block m slips off block M, we need to compare the maximum static friction force between the blocks to the force exerted due to the spring.

First, let's find the force exerted by the spring:

The force exerted by the spring, F_spring, is given by Hooke's Law:

F_spring = k * x

where k is the spring constant (130 N/m) and x is the displacement (0.10 m).

F_spring = (130 N/m) * (0.10 m)
F_spring = 13 N

Next, let's calculate the maximum static friction force between the blocks:

The maximum static friction force, F_static max, is given by:

F_static max = u_static * N

where u_static is the coefficient of static friction (0.3) and N is the normal force between the blocks.

The normal force can be calculated using the masses and the acceleration due to gravity (9.8 m/s^2):

N = (M + m) * g

where M is the mass of the bottom block (5.00 kg), m is the mass of the top block (1.25 kg), and g is the acceleration due to gravity (9.8 m/s^2).

N = (5.00 kg + 1.25 kg) * 9.8 m/s^2
N = 61.25 N

Now, let's calculate the maximum static friction force:

F_static max = (0.3) * (61.25 N)
F_static max = 18.375 N

Finally, let's compare the maximum static friction force to the force exerted by the spring:

Since F_spring (13 N) is less than F_static max (18.375 N), the blocks will not slip off each other.

Therefore, the answer is NO, block m does not slip off block M.