# An object, previously at rest, slides #9 m# down a ramp, with an incline of #(pi)/3 #, and then slides horizontally on the floor for another #12 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

The frictional coefficient of the surface is given by

Frictional coefficients have no units.

While traveling, the object accelerates on the inclined plane and decelerates on the floor. It starts and ends its journey at rest.

\sigh} Will try again tomorrow to finish it up; I'm tired and having trouble with this one.

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To solve for the material's kinetic friction coefficient, we need to analyze the motion of the object and apply principles of energy conservation.

The total work done on the object equals the change in its kinetic energy. This can be expressed as the work done against friction on the ramp plus the work done against friction on the horizontal floor.

The work done against friction on the ramp can be calculated using the gravitational potential energy the object loses as it descends the ramp. The work done against friction on the horizontal floor is equal to the frictional force multiplied by the distance.

Let's denote the kinetic friction coefficient as ( \mu_k ).

The gravitational potential energy lost on the ramp is converted into kinetic energy and work against friction:

( mgh = \frac{1}{2}mv_r^2 + f_r d_r ),

where ( m ) is the mass of the object, ( g ) is the acceleration due to gravity, ( h ) is the vertical height of the ramp, ( v_r ) is the velocity of the object on the ramp, ( f_r ) is the frictional force on the ramp, and ( d_r ) is the distance down the ramp.

The work done on the horizontal floor is:

( f_h d_h ),

where ( f_h ) is the frictional force on the horizontal floor, and ( d_h ) is the horizontal distance.

Given that the object slides down the ramp, its vertical height is ( h = 9 \sin(\pi/3) ). The velocity on the ramp, ( v_r ), can be found using the equations of motion. Since the object started from rest, we can use ( v_r^2 = 2gh ).

Now, since the material is the same, the coefficient of friction is consistent throughout, so ( f_r = f_h = \mu_k mg ).

By substituting the known values and solving for ( \mu_k ), we can find the kinetic friction coefficient.

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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.

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- If the length of a #42 cm# spring increases to #87 cm# when a #1 kg# weight is hanging from it, what is the spring's constant?
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