A 2.8 kg mass hangs at one end of a rope that is attached to a support on a railroad car. When the car accelerates to the right, the rope makes an angle of 9.0° with the vertical, as shown in Figure P4.68. Find the acceleration of the car.
tan(9)*9.8
To find the acceleration of the car, we need to analyze the forces acting on the mass.
Let's break down the forces:
1. Gravitational Force (Weight): The weight (W) of the mass can be calculated using the formula W = mg, where m is the mass (2.8 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
W = 2.8 kg * 9.8 m/s^2 ≈ 27.44 N
This gravitational force acts vertically downward.
2. Tension in the Rope: The tension (T) in the rope is the force that keeps the mass suspended. It can be broken down into horizontal (T_H) and vertical (T_V) components.
Since the rope makes an angle of 9.0° with the vertical, we can find the vertical component using T_V = T * cos(9.0°) and the horizontal component using T_H = T * sin(9.0°).
3. Net Force: The net force acting on the mass is the sum of the horizontal forces. Since the system is accelerating to the right, the horizontal net force is responsible for this acceleration.
The horizontal net force is given by F_net = T_H.
4. Equating Forces: To find the acceleration of the car, we equate the net force with the mass of the object multiplied by its acceleration:
F_net = m * a, where a is the acceleration of the car.
Therefore, we have T_H = m * a.
Using the components of T, we can express T_H in terms of T:
T_H = T * sin(9.0°).
Substituting this into the equation above, we get:
T * sin(9.0°) = m * a.
Now, we can solve for a, which is the acceleration of the car:
a = T * sin(9.0°) / m.
To find T, we need to consider the vertical forces acting on the mass:
T_V - W = 0,
since the vertical net force is zero (the mass is not moving up or down).
Substituting the values we found earlier:
T * cos(9.0°) - 27.44 N = 0.
We can solve for T using this equation:
T = 27.44 N / cos(9.0°).
Finally, substituting T into the equation for the acceleration, we get:
a = (27.44 N / cos(9.0°)) * sin(9.0°) / m.
Evaluating this expression will give us the acceleration of the car.