A 2.6-kg object 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 7.5° with the vertical, as shown in the figure below. Find the acceleration of the car
To find the acceleration of the car, we need to analyze the forces acting on the hanging object.
1. First, let's draw a free-body diagram of the object. We have the weight (mg) acting vertically downward and the tension force (T) acting along the rope.
2. The weight (mg) can be divided into two components: one parallel to the rope (mg * sinθ) and one perpendicular to the rope (mg * cosθ), where θ is the angle made by the rope with the vertical.
3. The net force acting parallel to the surface of the car is responsible for the acceleration of the car. In this case, it is the horizontal component of tension (T * sinθ) minus the component of the weight (mg * sinθ).
4. Now, we can set up the equation using Newton's second law of motion: F = ma, where F is the net force, m is the mass of the object, and a is the acceleration of the car.
T * sinθ - mg * sinθ = ma
5. We know the mass of the object (2.6 kg) and the angle made by the rope with the vertical (θ = 7.5°). We also have the acceleration of gravity (g ≈ 9.8 m/s²).
6. Plugging in the values, we can solve for the acceleration of the car:
T * sinθ - mg * sinθ = ma
T * sin(7.5°) - (2.6 kg * 9.8 m/s²) * sin(7.5°) = 2.6 kg * a
T * 0.1316 - (25.48 kg·m/s²) * 0.1316 = 2.6 kg * a
T * 0.1316 - 3.35887 kg·m/s² = 2.6 kg * a
7. Since we know the tension force (T) is equal to the weight of the object (mg), we can simplify the equation further:
(2.6 kg * 9.8 m/s²) * 0.1316 - 3.35887 kg·m/s² = 2.6 kg * a
25.48 kg·m/s² * 0.1316 - 3.35887 kg·m/s² = 2.6 kg * a
8. Solving this equation will give us the acceleration of the car (a).