# An object with a mass of #6 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 2+cscx #. How much work would it take to move the object over #x in [pi/8, (3pi)/4], where x is in meters?

The work is

The work done is

The frictional force is

The work done is

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To calculate the work done to move the object over the given interval, you need to integrate the force of kinetic friction with respect to distance ( x ) over the interval ( [\frac{\pi}{8}, \frac{3\pi}{4}] ).

The force of kinetic friction is given by ( f_k(x) = \mu_k(x) \cdot N ), where ( \mu_k(x) = 2 + \csc(x) ) and ( N ) is the normal force.

Given that the object is pushed along a linear path, the normal force is constant. Let's denote it as ( N_0 ).

Therefore, the work done is given by the integral:

[ W = \int_{\frac{\pi}{8}}^{\frac{3\pi}{4}} f_k(x) , dx ]

[ W = \int_{\frac{\pi}{8}}^{\frac{3\pi}{4}} (2 + \csc(x)) \cdot N_0 , dx ]

You'll need to evaluate this integral to find the exact amount of work done.

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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 #10 kg# is on a plane with an incline of # - pi/4 #. If it takes #12 N# to start pushing the object down the plane and #4 N# to keep pushing it, what are the coefficients of static and kinetic friction?
- If an object with a mass of #10 kg # is moving on a surface at #15 ms^-1# and slows to a halt after # 4 s#, what is the friction coefficient of the surface?
- An object, previously at rest, slides #5 m# down a ramp, with an incline of #(3pi)/8 #, and then slides horizontally on the floor for another #15 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?
- An object with a mass of #5 # # kg# is on a surface with a kinetic friction coefficient of # 4 #. How much force is necessary to accelerate the object horizontally at # 6# #ms^-2#?
- If an object with a mass of #10 kg # is moving on a surface at #45 m/s# and slows to a halt after # 6 s#, what is the friction coefficient of the surface?

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