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.
ma = Tsinθ
mg= Tcosθ
a/g =tanθ =>
a=g• tanθ
Hey can you please tell the numrical answer I can`t get it :)
2.626
2.626
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To determine the acceleration of your car, we need to use the given information about the angle of the fuzzy dice and the acceleration due to gravity.
When the car accelerates, the fuzzy dice will experience two forces: the force of gravity pulling them downward and the force of acceleration pushing them in the direction of the car's motion. These forces result in the fuzzy dice hanging at an angle.
The force of gravity acting on the fuzzy dice can be calculated using the equation:
F_gravity = m * g
Where m is the mass of the fuzzy dice and g is the acceleration due to gravity (-9.8 m/s²).
Since the fuzzy dice are hanging at an angle, the vertical component of the force of gravity balances out the vertical component of the force of acceleration. This can be demonstrated by drawing a free body diagram of the fuzzy dice. The remaining horizontal component of the force of acceleration can be calculated using trigonometry:
F_horizontal = F_acceleration * cos(θ)
Where θ is the angle of 15 degrees.
Since the force of acceleration is resulting from the car's acceleration, we can equate this horizontal force with the mass of the fuzzy dice multiplied by the car's acceleration:
F_horizontal = m * a
Combining the equations, we can solve for the car's acceleration (a):
m * a = F_acceleration * cos(θ)
a = F_acceleration * cos(θ) / m
Now, substituting the equation for the force of acceleration (F_acceleration) into the above equation, we get:
a = (m * g * cos(θ)) / m
The mass cancels out, leaving us with:
a = g * cos(θ)
Substituting the values given:
a = (-9.8 m/s²) * cos(15°)
Calculating this expression gives us the acceleration of your car in m/s².