A ball hands on a string in the bus from the rear-view mirror. Once the driver hits the gas pedal, and while he accelerates, the ball no longer hangs straight down, but instead make an angle è of 15 degrees with respect to the vertical. How fast is the bus driver accelerating in m/s^2 ?
To determine the acceleration of the bus driver, we need to analyze the forces acting on the ball.
When the bus is at rest or moving at a constant velocity, the ball hangs straight down because the forces acting on it are balanced. The downward force due to gravity is balanced by the tension force in the string.
However, when the bus starts to accelerate, an additional force acts on the ball in the horizontal direction. This force is the result of the inertia of the ball, which resists changes in its motion. As a result, the ball is pushed backward by the acceleration of the bus.
To find the acceleration of the bus driver, we can apply trigonometry to the situation. We know that the angle (θ) between the string and the vertical is 15 degrees.
The force due to gravity acting on the ball can be split into two components: one in the vertical direction (mg) and one in the horizontal direction (0). The force resisting the ball's motion in the horizontal direction is equal to the tension in the string (T) and is responsible for the ball's angle with respect to the vertical.
Since the forces are perpendicular, we can use trigonometry to relate the angle (θ) and the forces involved:
tanθ = (mg) / T
Rearranging the equation, we have:
T = (mg) / tanθ
Now, we can consider the net force acting on the ball in the horizontal direction:
F_net = T - ma
Where m is the mass of the ball and a is the acceleration of the bus driver.
Since the ball is not moving vertically, the net force in the vertical direction is zero:
F_net_vertical = T - mg = 0
Solving for T, we get:
T = mg
Substituting T = mg into the net force equation, we have:
mg = ma
Canceling the mass, we find:
g = a
Therefore, the acceleration of the bus driver is equal to the acceleration due to gravity (9.8 m/s²).