A pair of fuzzy dice are hanging by a string in my car from the rear-view mirror. I hit the gas pedal, and while I accelerate the fuzzy dice no longer hang straight down, but instead make an angle θ of 15 degrees with respect to the vertical. How fast am I accelerating in m/s2?
Details and assumptions
The acceleration of gravity is −9.8 m/s2.
Treat the fuzzy dice as point masses.
2.62
To find the acceleration, we can analyze the forces acting on the fuzzy dice.
When the car accelerates, there are two forces acting on the fuzzy dice: the gravitational force pulling them downward and the net force due to the acceleration pushing them in the opposite direction.
Let's break down the forces:
1. Gravitational force (Fg): This force is given by the mass of the fuzzy dice (which we'll assume is m) multiplied by the acceleration due to gravity (-9.8 m/s^2). Therefore, the gravitational force is Fg = -9.8m.
2. Net force due to acceleration (Fa): This force is equal to the mass of the fuzzy dice multiplied by the acceleration of the car (which we'll assume is a). Thus, the net force due to acceleration is Fa = ma.
Considering the angles involved, we can divide the gravitational force and the net force due to acceleration into their vertical and horizontal components:
Vertical components:
- For the gravitational force (Fg), the vertical component is given by Fg_vertical = -9.8m * cos(θ), where θ is the angle of 15 degrees. Since the fuzzy dice hang at an angle, this vertical component counteracts the gravitational force.
- For the net force due to acceleration (Fa), the vertical component is given by Fa_vertical = ma * sin(θ), where θ is again the angle of 15 degrees. The vertical component of the net force due to acceleration adds to the gravitational force.
Setting these two vertical components equal to each other, we have: -9.8m * cos(θ) = ma * sin(θ).
Simplifying, we find the acceleration of the car (a): a = -9.8 * cos(θ) / sin(θ).
Given that θ is 15 degrees, we can plug in the values and calculate:
a = -9.8 * cos(15°) / sin(15°).
Using a calculator, we find:
a ≈ -4.67 m/s^2.
Therefore, the car is accelerating at approximately -4.67 m/s^2.