An object with a mass of #6 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 1+3cscx #. How much work would it take to move the object over #x in [(pi)/6, (pi)/4], where x is in meters?

Answer 1
#w=int_(pi/6) ^(pi/4)mu_k(x)*m*gdx# #w=mgint_(pi/6) ^(pi/4)(1+3cscx)dx# #w=mgint_(pi/6) ^(pi/4)dx+3mgint_(pi/6) ^(pi/4)cscxdx#
#=mgint_(pi/6) ^(pi/4)dx+3mgint_(pi/6) ^(pi/4)cscxdx#
#= mg[(pi/4-pi/6)](pi/6)-3mgln(csc(pi/4)+cot(pi/4)-ln(csc(pi/6)+cot(pi/6)))# pl put m=6kg #g=9.8ms^-2# and values from trigonometrical table and calculate #w= 6xx9.8*pi/12-3xx6xx9.8xxln((sqrt2+1)/(2+sqrt3))J#
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Answer 2

To find the work done, integrate the force of kinetic friction over the given interval. The force of kinetic friction ( F_k ) is given by ( F_k = \mu_k \cdot N ), where ( \mu_k ) is the kinetic friction coefficient and ( N ) is the normal force. The normal force ( N ) equals the weight of the object, which is ( mg ), where ( m ) is the mass and ( g ) is the acceleration due to gravity. Substitute ( N ) and ( \mu_k ) into the equation for ( F_k ) and integrate ( F_k ) over the given interval to find the work done. The work done ( W ) is equal to the negative of the integral of ( F_k ) with respect to ( x ) over the given interval. So, ( W = -\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} (\mu_k(x) \cdot mg) , dx ).

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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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