A car is climbing a hill and accelerating at a constant rate of 0.3 g as it does so. Φ=10°. What is the steady state value of θ, the angle made by a mass hung from a cord that’s attached to the rearview mirror.
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arctan(a/g) is added to 10deg
force parallel to slope = .3 g m + m g sin 10 down slope = .474 m g
force normal to slope = m g cos 10 =.985 m g
tan angle relative to normal to slope = .474/.985 = .481
so
angle relative to normal to slope (straight down in the car) = 25.7 degrees or 15.7 degrees from direction to earth center
To find the steady state value of θ, we need to analyze the forces acting on the mass hung from the cord attached to the rearview mirror.
Let's break down the forces:
1. Weight force (Fg): The weight of the mass acts vertically downward, given by Fg = m * g, where m represents the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Tension force (T): The tension force is acting along the cord, keeping the mass in its position. The tension force can be decomposed into horizontal (Tx) and vertical (Ty) components.
3. Centripetal force (Fc): The centripetal force is acting towards the center of the circular motion of the mass. This force is created by the acceleration of the car and can be calculated using Fc = m * a, where a is the acceleration of the car.
Given that the car is accelerating at a constant rate of 0.3 g, the centripetal force can be calculated as Fc = m * (0.3 * g).
Next, we can break down the tension force:
- The vertical component of the tension force is balancing out the weight force, so Ty = Fg.
- The horizontal component of the tension force is balancing out the centripetal force, so Tx = Fc.
Now, to find the angle θ, we can use the trigonometric relationship:
tan(θ) = Ty / Tx.
Substituting the values:
tan(θ) = Fg / Fc.
Substituting Fg = m * g and Fc = m * (0.3 * g):
tan(θ) = g / (0.3 * g).
Simplifying:
tan(θ) = 1 / 0.3.
Using the inverse tangent function (arctan) to solve for θ:
θ = arctan(1 / 0.3).
Evaluating this using a calculator:
θ ≈ 73.74°.
Therefore, the steady state value of θ, the angle made by the mass hung from the cord that's attached to the rearview mirror, is approximately 73.74°.
To find the steady state value of θ, we need to understand the forces acting on the mass hung from the cord.
First, let's look at the forces acting on the mass:
1. Gravitational Force: The mass hangs vertically downward, so it experiences a force equal to its weight, which is given by F_gravity = m * g, where m is the mass and g is the acceleration due to gravity.
2. Tension Force: The cord attached to the rearview mirror applies a tension force on the mass, which provides the necessary centripetal force to keep the mass moving in a circle. This tension force can be broken down into two components:
a. Horizontal Component: This component provides the centripetal force and causes the mass to move in a circle. It is given by F_horizontal = m * a_c, where a_c is the centripetal acceleration. In this case, the centripetal acceleration is provided by the acceleration of the car.
b. Vertical Component: This component acts in the upward direction and opposes the gravitational force. It is given by F_vertical = m * g * sin(θ), where θ is the angle made by the mass.
Now, based on the given information, the car is climbing a hill with an acceleration of 0.3 g. This means the acceleration of the car is 0.3 times the acceleration due to gravity. So, a_c = 0.3 * g.
To find the steady state value of θ, we need to consider the conditions where the vertical component of the tension force balances the gravitational force.
Setting the vertical forces equal:
m * g * sin(θ) = m * g
Canceling the mass:
sin(θ) = 1
From this equation, we can find the value of θ that satisfies this condition. We can use an inverse trigonometric function to find the angle.
θ = arcsin(1)
Asin is the inverse of sin, so the equation simplifies to:
θ = sin^(-1)(1)
Now, evaluating this equation, we find:
θ ≈ 90°
Therefore, the steady state value of θ, the angle made by the mass hung from the cord, is approximately 90°.