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?

Answer 1

The work is #=98.0J#

We need

#intcotxdx=ln|sin(x)|+C#

The work done is

#W=F*d#

The frictional force is

#F_r=mu_k*N#
The normal force is #N=mg#
The mass is #m=6kg#
#F_r=mu_k*mg#
#=6*(1+cotx)g#

The work done is

#W=6gint_(1/8pi)^(3/8pi)(1+cotx)dx#
#=6g*[x+ln|sin(x)|]_(1/8pi)^(3/8pi)#
#=6g(3/8pi+ln|sin(3/8pi)|-1/8pi-ln|sin(1/8pi)|)#
#=6g(1.667)#
#=98.0J#
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Answer 2

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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Answer from HIX Tutor

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.

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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