Question

problem 1: a ball rolls down a ramp from an initial position of (0,10) m to a final position of (3,1) m. find the work done by gravity. does the path taken matter? note that the gravitational acceleration is (0, -9.8) m/s²

Explanation:

Step1: Find the displacement vector

The initial position vector $\vec{r}_i=(0, 10)$ and the final position vector $\vec{r}_f=(3,1)$. The displacement vector $\vec{d}=\vec{r}_f-\vec{r}_i=(3 - 0,1 - 10)=(3,-9)$.

Step2: Find the gravitational force vector

The gravitational force vector $\vec{F}=m\vec{g}$, where $\vec{g}=(0,-9.8)$. Let the mass of the ball be $m$. So $\vec{F}=(0,-9.8m)$.

Step3: Calculate the work - done by gravity

The work - done $W=\vec{F}\cdot\vec{d}$. Using the dot - product formula $\vec{a}\cdot\vec{b}=a_xb_x + a_yb_y$, we have $W=(0\times3)+(-9.8m\times(-9)) = 88.2m$ joules.

Step4: Determine if the path matters

Gravitational force is a conservative force. For a conservative force, the work done between two points is path - independent. So the path taken does not matter.

Answer:

The work done by gravity is $88.2m$ joules and the path taken does not matter.