A 0.0820 kg pair of fuzzy dice is attached to the rearview mirror of a car by a short string. The car accelerates at a constant rate, and the fuzzy dice hang at an angle due to the car's acceleration. If the dice hang at 10.5° from the vertical as the car accelerates, what is the acceleration of the car?
To find the acceleration of the car, we can use the formula for the net force acting on the fuzzy dice in the vertical direction.
The net force in the vertical direction is the tension force in the string minus the force due to gravity.
Let's break down the forces:
1. Force due to gravity (Fg): The force due to gravity is equal to the weight of the fuzzy dice, which is its mass multiplied by the acceleration due to gravity (9.8 m/s^2).
Fg = (mass of fuzzy dice) * (acceleration due to gravity)
2. Tension force (T): The tension force in the string is the force that counteracts the force due to gravity.
T = Fg (to keep the dice hanging at an angle of 10.5° from the vertical)
Now, let's find the acceleration of the car:
Step 1: Calculate the force due to gravity:
Fg = (mass of fuzzy dice) * (acceleration due to gravity)
Fg = 0.0820 kg * 9.8 m/s^2
Fg = 0.8036 N
Step 2: Calculate the tension force:
T = Fg (since the dice hang at an angle of 10.5° from the vertical)
T = 0.8036 N
Step 3: Use Newton's second law to find the net force in the vertical direction:
Net force (Fnet) = T - Fg (since the net force is the tension force minus the force due to gravity, both acting in opposite directions)
Fnet = 0.8036 N - 0.8036 N
Fnet = 0 N
Step 4: Use the net force and the mass of the dice to find the acceleration of the car:
Fnet = mass of dice * acceleration of the car (in the vertical direction)
0 N = 0.0820 kg * acceleration of the car
acceleration of the car = 0 N / 0.0820 kg
acceleration of the car = 0 m/s^2
Therefore, the acceleration of the car is 0 m/s^2.
To determine the acceleration of the car, we need to analyze the forces acting on the fuzzy dice.
First, we can consider the forces in the vertical direction. The weight of the dice, acting downward, can be calculated using the formula: weight = mass * gravity. As the dice have a mass of 0.0820 kg, and the acceleration due to gravity is approximately 9.8 m/s^2, the weight of the dice is 0.0820 kg * 9.8 m/s^2 = 0.8036 N.
Next, we can consider the forces in the horizontal direction. The only force acting on the dice in this direction is the tension in the string. Since the dice are hanging at an angle of 10.5° from the vertical, the tension T in the string can be determined using the formula: T = weight / sin(angle). Plugging in the values, we have: T = 0.8036 N / sin(10.5°) = 0.8036 N / 0.1821 = 4.412 N.
Now, we can relate the tension in the string to the acceleration of the car. The tension T in the string can be related to the mass m of the dice and the acceleration a of the car using the formula: T = m * a. Rearranging the formula, we have: a = T / m = 4.412 N / 0.0820 kg = 53.8549 m/s^2.
Therefore, the acceleration of the car is approximately 53.8549 m/s^2.