A block with mass m 1 = 9.3 kg is on an incline with an angle ? = 40 with respec

Question

A block with mass m1 = 9.3 kg is on an incline with an angle ? = 40 with respect to the horizontal. For the first question there is no friction, but for the rest of this problem the coefficients of friction are: ?k = 0.38 and ?s = 0.418.

1) When there is no friction, what is the magnitude of the acceleration of the block?   m/s2

2) Now with friction, what is the magnitude of the acceleration of the block after it begins to slide down the plane? m/s2

3)

To keep the mass from accelerating, a spring is attached. What is the minimum spring constant of the spring to keep the block from sliding if it extends x = 0.12 m from its unstretched length.   N/m

4)

Now a new block with mass m2 = 14.5 kg is attached to the first block. The new block is made of a different material and has a greater coefficient of static friction. What minimum value for the coefficient of static friction is needed between the new block and the plane to keep the system from accelerating?

Explanation / Answer

If you rotate the problem to where normal force is perpendicular to a axis, you will have a right triangle whose gravitational component is going southwest of the x axis

-----------------------mass-----
''
' "
' "
' "
'---------"

When there is no friction a=g,

f=ma = 9.3*9.8 = 91.14 N this is the force for the gravity component of the triangle. SW 91.14cos60=adj= 45.57N

45.57N/9.3= 4.9m/s^2

4)By F(net) = F(gravity) - [F(friction) + F(by spring)]
For F(net) =0
=>F(gravity) = F(friction) + F(by spring)
=>mgsin40* = ?s x mgcos40* + kx
=>9.3 x 9.8 x 0.63 = 0.418 x 9.5 x 9.8 x 0.78 + k x 0.12
=>k = 28.768/0.12 = 239.74 N/m

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