How much work was done on the book between A and B?
Understand the Problem
The question is asking for the calculation of the work done on a book as it slides between two points, A and B, considering its change in speed due to friction on a rough surface.
Answer
The work done on the book between points A and B is approximately $-6.556 \, \text{J}$.
Answer for screen readers
The work done on the book between points A and B is approximately $-6.556 , \text{J}$.
Steps to Solve
- Identify the given information
We know the mass of the book $m = 1.50 , \text{kg}$, the speed at point A $v_A = 3.21 , \text{m/s}$, and the speed at point B $v_B = 1.25 , \text{m/s}$.
- Calculate the initial and final kinetic energy
The kinetic energy at point A can be calculated using the formula:
$$ KE_A = \frac{1}{2} m v_A^2 $$ Plugging in the values:
$$ KE_A = \frac{1}{2} (1.50) (3.21)^2 $$
Now perform the calculation:
$$ KE_A = \frac{1}{2} (1.50) (10.3041) = 7.7281 , \text{J} $$
Next, calculate the kinetic energy at point B:
$$ KE_B = \frac{1}{2} m v_B^2 $$ Plugging in the values:
$$ KE_B = \frac{1}{2} (1.50) (1.25)^2 $$
Now perform the calculation:
$$ KE_B = \frac{1}{2} (1.50) (1.5625) = 1.171875 , \text{J} $$
- Calculate the change in kinetic energy
The work done on the book is equal to the change in kinetic energy:
$$ W = KE_B - KE_A $$ Substituting the values:
$$ W = 1.171875 - 7.7281 $$
Now perform the subtraction:
$$ W = -6.556225 , \text{J} $$
- Round the answer
Finally, round the answer to three decimal places:
$$ W \approx -6.556 , \text{J} $$
The work done on the book between points A and B is approximately $-6.556 , \text{J}$.
More Information
This negative value indicates that work was done against the motion of the book, primarily due to friction.
Tips
- Forgetting to square the velocity: Make sure to square the velocity when calculating kinetic energy.
- Incorrect sign of work: Remember that if energy is lost, the work done is negative.
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