A fuzzy die (m = 0.2 kg) hangs from the ceiling of a car. When the car accelerates forward along a horizontal road, the die swings backward so the the string supporting it makes an angle of 4o with the vertical. What is the acceleration of the car?
Well, well, looks like our fuzzy little friend is getting a swinging sensation in the car! Let's find out what's causing all the excitement!
Now, the angle that the string supporting the fuzzy die makes with the vertical is 4 degrees. So, we know that the net force acting on the die must be towards the rear of the car in order to make it swing backward.
To find the acceleration of the car, we need to find the net force acting on the die. And to do that, we need to know the gravitational force acting on the die.
The gravitational force is given by the equation F_gravity = m * g, where m is the mass of the fuzzy die and g is acceleration due to gravity.
Since the die is hanging from the ceiling, the net force acting on it must be equal to the component of the gravitational force along the direction of the string. This force can be given by the equation F_net = F_gravity * sin(theta), where theta is the angle the string makes with the vertical.
So, we have F_net = m * g * sin(theta).
Since the net force is also equal to the mass of the die multiplied by its acceleration, we can write m * a = m * g * sin(theta).
Therefore, the acceleration of the car, a, is given by the equation a = g * sin(theta).
Now, plug in the values: g is approximately 9.8 m/s^2, and theta is 4 degrees (or 0.07 radians).
Calculating a = 9.8 m/s^2 * sin(0.07) ≈ 0.7 m/s^2.
So, the acceleration of the car is approximately 0.7 m/s^2. Hope that puts a swing in your step!
To find the acceleration of the car, we can use the relationship between the angle at which the string supporting the die makes with the vertical and the acceleration of the car.
Let's start by drawing a free-body diagram of the forces acting on the die when it swings backward:
1. The weight of the die acts vertically downward. It can be resolved into two components:
- The component perpendicular to the string, which balances the tension in the string.
- The component parallel to the string, which causes the acceleration of the die.
2. The tension in the string acts along the string, away from the die.
Based on the given information, we know that the angle between the string and the vertical is 4 degrees.
Now, let's analyze the forces acting on the die:
1. The weight of the die can be expressed as: mg, where m is the mass of the die (0.2 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²).
2. The component of the weight perpendicular to the string can be expressed as: mg * cosθ, where θ is the angle between the string and the vertical (4 degrees).
3. The tension in the string can be expressed as: T.
Since the component of the weight perpendicular to the string balances the tension in the string, we can write an equation:
mg * cosθ = T
Now, let's look at the component of the weight parallel to the string:
1. The component of the weight parallel to the string can be expressed as: mg * sinθ, where θ is the angle between the string and the vertical (4 degrees).
2. This component of the weight causes the acceleration of the die. The magnitude of this acceleration is given by: a = mg * sinθ / m, where m is the mass of the die (0.2 kg).
Substituting the values into the equation:
a = (0.2 kg) * (9.8 m/s²) * sin(4 degrees) / (0.2 kg)
Now, we can calculate the acceleration of the car.
To find the acceleration of the car, we can start by analyzing the forces acting on the fuzzy die.
The only two forces that act on the die are its weight (mg) and the tension in the string (T). In this case, the weight acts vertically downward, while the tension in the string acts at an angle.
Given that the string makes an angle of 4 degrees with the vertical, we can break down the tension force into two components: one parallel to the vertical direction (T⊥) and the other parallel to the horizontal direction (T∥).
Since the die is in equilibrium (i.e., not moving up or down), the sum of the vertical components of the forces must be zero. Therefore:
T⊥ - mg = 0
Similarly, the sum of the horizontal components of the forces must be zero since the die does not move horizontally:
T∥ = 0
From the information provided, the angle between the string and the vertical is 4 degrees. Therefore, we can write:
tan(4) = T∥ / T⊥
Now, we can substitute the second equation into the first equation to solve for T⊥:
T⊥ - mg = 0
T⊥ = mg
Substituting the value of T⊥ into the second equation, we get:
tan(4) = T∥ / mg
Now, solving for T∥, we have:
T∥ = mg * tan(4)
The tension force parallel to the horizontal direction (T∥) is also equal to the net force acting on the die in the horizontal direction (F∥). And since we know that force is equal to mass times the acceleration (F = ma), we can write:
T∥ = ma
Equating the two equations for T∥, we have:
ma = mg * tan(4)
In this equation, m is the mass of the die, and a is the acceleration of the car. We want to solve for a, so we can rearrange the equation:
a = (mg * tan(4)) / m
Now, we can plug in the given values:
m = 0.2 kg
g = 9.8 m/s²
angle = 4 degrees
Converting the angle to radians, we have:
angle = 4 * π / 180 = 0.07 radians
Substituting the values into the equation, we get:
a = (0.2 kg * 9.8 m/s² * tan(0.07 radians)) / 0.2 kg
Simplifying the equation, we find that the acceleration of the car is:
a ≈ 0.14 m/s²
Therefore, the acceleration of the car is approximately 0.14 m/s².