problem 2: you push your desk horizontally in a circular arc of radius r = 3 m, as shown on the figure. the…

problem 2: you push your desk horizontally in a circular arc of radius r = 3 m, as shown on the figure. the mass of the desk is m = 15 kg, and the coefficient of kinetic friction between the floor and the desk is μk = 0.3. find the work done by friction as you go from a to b. would the answer change if the path you took was a straight line?

problem 2: you push your desk horizontally in a circular arc of radius r = 3 m, as shown on the figure. the mass of the desk is m = 15 kg, and the coefficient of kinetic friction between the floor and the desk is μk = 0.3. find the work done by friction as you go from a to b. would the answer change if the path you took was a straight line?

Answer

Explanation:

Step1: Calculate the normal force

The normal force $N$ on a horizontal - surface is equal to the weight of the object. Since $F_g=mg$ and $N = F_g$, where $m = 15\ kg$ and $g=9.8\ m/s^2$, we have $N=mg=15\times9.8 = 147\ N$.

Step2: Calculate the frictional force

The frictional force is given by $F_f=\mu_kN$. Substituting $\mu_k = 0.3$ and $N = 147\ N$, we get $F_f=0.3\times147=44.1\ N$.

Step3: Calculate the distance along the circular arc

The length of a semi - circular arc is $s=\pi R$, with $R = 3\ m$, so $s = 3\pi\ m$.

Step4: Calculate the work done by friction along the circular arc

The work done by a force $W = F\cdot d\cdot\cos\theta$. Since the frictional force is always opposite to the direction of motion, $\theta = 180^{\circ}$ and $\cos\theta=- 1$. So $W_{arc}=-F_fs=-44.1\times3\pi\approx - 415.5\ J$.

Step5: Calculate the work done by friction along a straight line

The distance along a straight line (diameter) between $A$ and $B$ is $d = 2R=6\ m$. The work done by friction is $W_{line}=-F_fd=-44.1\times6=-264.6\ J$.

Answer:

The work done by friction along the circular arc is approximately $-415.5\ J$. The answer would change if the path was a straight line, and the work done by friction along the straight - line path is $-264.6\ J$.