An object with a mass of #6 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 1+5cotx #. How much work would it take to move the object over #x in [(pi)/12, (3pi)/8]#, where #x# is in meters?

Question

An object with a mass of #6 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 1+5cotx   #.  How much work would it take to move the object over #x in [(pi)/12, (3pi)/8]#, where #x# is in meters?

Nathan Ashcroft · Accepted Answer

The work is #=428.1J#

#"Reminder : "#

#intcotxdx=ln(|sinx|)+C#

The work done is

#W=F*d#

The frictional force is

#F_r=mu_k*N#

The coefficient of kinetic friction is #mu_k=(1+5cotx)#

The normal force is #N=mg#

The mass of the object is #m=6kg#

#F_r=mu_k*mg#

#=6*(1+5cotx)g#

The work done is

#W=6gint_(1/12pi)^(3/8pi)(1+5cot)dx#

#=6g*[x+5ln(sin(x) ] _(1/12pi)^(3/8pi)#

#=6g((3/8pi+5ln(sin(3/8pi))-(1/12pi+5ln(sin(1/12pi))#

#=6xx9.8xx7.28#

#=428.1J#

Charlotte Aaron · Answer

undefined To find the work done in moving the object over the given interval, we need to integrate the product of the force of kinetic friction and the distance over that interval.

The force of kinetic friction $ F_k $ is given by $ 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 $ is equal to the weight of the object, which is $ m \cdot g $, where $ m $ is the mass of the object and $ g $ is the acceleration due to gravity.

Given $ m = 6 $ kg, $ g \approx 9.8 $ m/s$^2$, and the interval for $ x $ is $ \left[\frac{\pi}{12}, \frac{3\pi}{8}\right] $, we can calculate the work done using the following steps:

1. Calculate the normal force $ N = m \cdot g $.
2. Calculate the force of kinetic friction $ F_k = \mu_k \cdot N $ using $ \mu_k(x) = 1 + 5 \cot(x) $.
3. Integrate the product of $ F_k $ and $ dx $ over the given interval.

$$
W = \int_{\frac{\pi}{12}}^{\frac{3\pi}{8}} F_k(x) \, dx
$$

After calculating $ W $ using the definite integral, you will find the work done to move the object over the specified interval.