# An object with a mass of #16 kg# is lying still on a surface and is compressing a horizontal spring by #7/8 m#. If the spring's constant is #15 (kg)/s^2#, what is the minimum value of the surface's coefficient of static friction?

The coeficient of friction is

The static friction coefficient is

Consequently,

The typical force is

The static friction coefficient is

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To find the minimum value of the coefficient of static friction, we need to consider the forces acting on the object when it is at rest. The force exerted by the spring, which is ( kx ), where ( k ) is the spring constant and ( x ) is the compression of the spring. The force due to gravity, ( mg ), where ( m ) is the mass of the object and ( g ) is the acceleration due to gravity. The maximum force of static friction is ( \mu_s N ), where ( \mu_s ) is the coefficient of static friction and ( N ) is the normal force. At the minimum value of ( \mu_s ), the static friction force balances the net force acting on the object. Therefore, we have:

[ kx + \mu_s N = mg ]

[ kx + \mu_s mg = mg ]

[ \mu_s = \frac{kx}{mg} ]

Substituting the given values:

[ \mu_s = \frac{(15, \text{kg}/\text{s}^2) \cdot (7/8, \text{m})}{(16, \text{kg}) \cdot (9.8, \text{m/s}^2)} ]

[ \mu_s \approx \frac{131.25}{156.8} ]

[ \mu_s \approx 0.8366 ]

Therefore, the minimum value of the surface's coefficient of static friction is approximately ( \mu_s = 0.8366 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- An object with a mass of #12 kg# is lying still on a surface and is compressing a horizontal spring by #1/3 m#. If the spring's constant is #4 (kg)/s^2#, what is the minimum value of the surface's coefficient of static friction?
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- A box with an initial speed of #1 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #3/4 # and an incline of #pi /8 #. How far along the ramp will the box go?
- A box with an initial speed of #4 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #1/6 # and an incline of #pi /3 #. How far along the ramp will the box go?
- An object with a mass of #4 kg# is acted on by two forces. The first is #F_1= < 6 N , -2 N># and the second is #F_2 = < -5 N, 3 N>#. What is the object's rate and direction of acceleration?

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