A car, starting from rest, accelerates down a hill reaching 30.0 m/s in 6.00 s. During this time, the driver notices that the fuzzy dice hanging by a string from the ceiling is perpendicular to the
ceiling. Find (a) the angle θ of the hill and (b) the tension in the string.
a)30.7 degrees
b)843N
To find the angle θ of the hill, we can use the formula for acceleration down an inclined plane.
(a) First, we need to find the acceleration of the car down the hill. We can use the formula:
acceleration = (final velocity - initial velocity) / time
Given that the car is starting from rest (initial velocity = 0 m/s), the final velocity = 30.0 m/s, and the time = 6.00 s, we can substitute these values into the formula:
acceleration = (30.0 m/s - 0 m/s) / 6.00 s = 5.00 m/s²
Next, we can use the formula for acceleration down an inclined plane:
acceleration = g * sin(θ)
where g is the acceleration due to gravity (9.8 m/s²). We can rearrange the formula to solve for the angle θ:
θ = sin^(-1)(acceleration / g)
Substituting the values we have:
θ = sin^(-1)(5.00 m/s² / 9.8 m/s²) ≈ 30.3°
Therefore, the angle θ of the hill is approximately 30.3°.
(b) To find the tension in the string, we need to consider the forces acting on the fuzzy dice.
At equilibrium, the net force on the fuzzy dice is zero. We have two forces acting on it: the force of gravity (mg) and the tension in the string (T).
The force of gravity can be broken down into two components, parallel and perpendicular to the ceiling. The parallel component (mg*sin(θ)) contributes to the acceleration down the hill, while the perpendicular component (mg*cos(θ)) cancels out the tension in the string.
Therefore, we can write the equation:
mg*cos(θ) = T
To solve for T, we need to know the mass of the fuzzy dice, m. However, this information is not provided in the question. Without the mass, we cannot determine the tension in the string, so we cannot answer part (b) of the question.
In summary, the angle θ of the hill is approximately 30.3°, but without the mass of the fuzzy dice, we cannot determine the tension in the string.