A 3.2-kg block is placed at the top of a rough incline that has a vertical height of 0.68 m and a friction coefficient of $\mu _ { k } = 0.4$ . The block is released (with a gentle push) and slides all the way down. Before it comes to rest at the bottom, the amount of thermal energy it produces is

Question

Quizplus · Accepted Answer

The thermal energy produced is equal to the work done by friction. The work done by friction can be calculated using the formula $W = \mu_k \cdot m \cdot g \cdot d$, where $m$ is the mass, $g$ is the acceleration due to gravity (9.8 m/s$^2$), $\mu_k$ is the coefficient of kinetic friction, and $d$ is the distance traveled down the incline. The distance $d$ can be found using the Pythagorean theorem, given the vertical height $h = 0.68$ m. Assuming the incline is a right triangle, $d = \frac{h}{\sin(\theta)}$, where $\theta$ is the angle of the incline. However, without the length of the incline directly or the angle, we can simplify the problem by recognizing that the work done by friction is equal to the loss in potential energy, which does not depend on the path taken but only on the initial and final positions. Therefore, the thermal energy produced is equal to the initial potential energy of the block, which is $mgh$. Substituting the given values: $3.2 \cdot 9.8 \cdot 0.68$, we get approximately 21.3 J, which rounds to 21 J, making option B correct.

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