# An object with a mass of #6 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 1+cotx #. How much work would it take to move the object over #x in [(pi)/8, (3pi)/8], where x is in meters?

The work is

We need

The work done is

The frictional force is

The work done is

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To find the work done to move the object over the given interval, we need to integrate the force of kinetic friction with respect to displacement over that interval. The force of kinetic friction ( F_k ) can be calculated using the formula ( F_k = \mu_k \cdot N ), where ( \mu_k ) is the coefficient of kinetic friction and ( N ) is the normal force.

The normal force ( N ) can be calculated using the equation ( N = mg ), where ( m ) is the mass of the object and ( g ) is the acceleration due to gravity.

Since the coefficient of kinetic friction is given as ( \mu_k(x) = 1 + \cot(x) ), we need to express it in terms of ( x ) in the given interval ([ \frac{\pi}{8}, \frac{3\pi}{8} ]). Then, we can integrate ( F_k ) over the interval to find the work done.

The work done ( W ) is given by the equation ( W = \int_{a}^{b} F_k(x) , dx ), where ( a ) and ( b ) are the limits of integration.

Therefore, to find the work done, we first need to calculate the normal force ( N ), then calculate the force of kinetic friction ( F_k ) using ( F_k = \mu_k \cdot N ), and finally integrate ( F_k ) over the given interval ([ \frac{\pi}{8}, \frac{3\pi}{8} ]) with respect to ( x ).

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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.

- A truck pulls boxes up an incline plane. The truck can exert a maximum force of #2,400 N#. If the plane's incline is #(2 pi )/3 # and the coefficient of friction is #5/8 #, what is the maximum mass that can be pulled up at one time?
- An object with a mass of #12 kg# is on a surface with a kinetic friction coefficient of # 2 #. How much force is necessary to accelerate the object horizontally at # 14 m/s^2#?
- An object with a mass of #14 kg# is acted on by two forces. The first is #F_1= < 5 N , 3 N># and the second is #F_2 = < 2 N, 9 N>#. What is the object's rate and direction of acceleration?
- A 50 kg skater pushed by a friend accelerates at 5 m/s. How much force did the friend apply?
- An object, previously at rest, slides #8 m# down a ramp, with an incline of #pi/4 #, and then slides horizontally on the floor for another #16 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

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