Your friend's 11.1g tassel hangs on a string from his rear-view mirror. When he accelerates from a stop light, the tassel deflects backward toward the rear of the car. If the tassel hangs at an angle of 6.89deg relative to the vertical, 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 tassel:
F_net = m * a
where F_net is the net force, m is the mass of the tassel, and a is the acceleration of the car.
We can break down the forces acting on the tassel into two components: the force due to gravity, and the force due to the acceleration of the car.
1. Force due to gravity:
The force due to gravity can be calculated using the formula:
F_gravity = m * g
where m is the mass of the tassel and g is the acceleration due to gravity (approximately 9.8 m/s^2).
We can find the vertical component of this force using trigonometry:
F_vertical = F_gravity * sin(angle)
where angle is the angle between the tassel and the vertical. Therefore:
F_vertical = (m * g) * sin(angle)
2. Force due to acceleration:
The force due to acceleration can be calculated using the formula:
F_acceleration = m * a
where m is the mass of the tassel and a is the acceleration of the car.
Since the tassel is deflecting backward, the force due to acceleration must be directed opposite to the angle of deflection. Thus, the horizontal component of this force can be calculated using trigonometry:
F_horizontal = F_acceleration * cos(angle)
where angle is the angle between the tassel and the vertical. Therefore:
F_horizontal = (m * a) * cos(angle)
Since the tassel is in equilibrium, the net force on the tassel is zero. Therefore, we have:
F_vertical = F_horizontal
Substituting the previously obtained expressions:
(m * g) * sin(angle) = (m * a) * cos(angle)
Simplifying:
g * sin(angle) = a * cos(angle)
Solving for a:
a = (g * sin(angle)) / cos(angle)
Plugging in the values:
a = (9.8 m/s^2 * sin(6.89 deg)) / cos(6.89 deg)
To find the acceleration of the car, we need to understand the forces acting on the tassel.
When the car accelerates, there are two forces acting on the tassel: the gravitational force (mg) pulling it downward, and the net force (ma) responsible for causing the tassel to deflect backward.
Let's break down these forces:
1. Gravitational Force (mg):
The weight of the tassel is given by the product of its mass (11.1g) and the acceleration due to gravity (9.8 m/s²):
Weight (mg) = (0.0111 kg) × (9.8 m/s²) = 0.10878 N
2. Net Force (ma):
The tassel deflects backward, so we need to consider the component of the net force acting in that direction. This component can be calculated using trigonometry:
Force component = Weight × sin(angle)
Force component = 0.10878 N × sin(6.89°)
Now, we know that the net force is equal to mass times acceleration (ma), so we can set up the equation:
ma = Force component
Since the mass of the tassel is given (0.0111 kg), we can rearrange the equation to solve for acceleration (a):
a = (Force component) / m
Substituting the values:
a = (0.10878 N × sin(6.89°)) / 0.0111 kg
Evaluating this expression will give us the acceleration of the car.