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 can use the concept of forces and their components. Let me explain step by step how to solve this problem:
Step 1: Draw a free-body diagram of the hanging object. In this case, the force of gravity (mg) acts vertically downward, and the tension in the rope (T) acts upwards at an angle of 7.5° with the vertical.
Step 2: Resolve the forces into their components. Since the only force acting vertically is the weight (mg), we can write:
∑Fy = T * cos θ - mg = 0
where ∑Fy represents the sum of the forces in the vertical direction, T is the tension in the rope, θ is the angle the rope makes with the vertical, and mg is the weight of the object.
Step 3: Resolve the forces in the horizontal direction. In this case, the only force is the horizontal component of the tension, which acts towards the right. We can write:
∑Fx = T * sin θ = m * a
where ∑Fx represents the sum of the forces in the horizontal direction, m is the mass of the object, and a is the acceleration of the car.
Step 4: Solving the equations. We can solve the first equation from Step 2 for T:
T = mg / cos θ
Substituting this into the second equation, we get:
(mg / cos θ) * sin θ = m * a
Simplifying the equation, we have:
m * g * tan θ = m * a
Finally, dividing both sides by the mass (m), we get:
g * tan θ = a
where g is the acceleration due to gravity, approximately 9.8 m/s².
Step 5: Calculate the acceleration. Now, we can substitute the given values into the equation. The mass of the object (m) is given as 2.6 kg, and the angle (θ) is given as 7.5°. The acceleration (a) can be calculated by multiplying the acceleration due to gravity (g) by the tangent of the angle:
a = g * tan θ
a = 9.8 m/s² * tan(7.5°)
Using a calculator, we find:
a ≈ 1.32 m/s²
Therefore, the acceleration of the car is approximately 1.32 m/s² to the right.
To find the acceleration of the car, we can use the tension in the rope and the weight of the object.
Step 1: Calculate the weight of the object.
The weight (W) of an object can be calculated using the formula:
W = m * g
where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s²).
Given:
Mass of the object (m) = 2.6 kg
Using the formula:
W = 2.6 kg * 9.8 m/s² = 25.48 N
Step 2: Calculate the vertical component of the tension force.
The tension in the rope consists of two components: a vertical component (T_v) and a horizontal component (T_h).
The vertical component of the tension (T_v) can be determined using the formula:
T_v = W * cos(theta)
where theta (θ) is the angle the rope makes with the vertical, which is 7.5° in this case.
Using the formula:
T_v = 25.48 N * cos(7.5°) = 25.48 N * 0.992546 = 25.30 N
Step 3: Calculate the horizontal component of the tension force.
The horizontal component of the tension (T_h) can be calculated using the formula:
T_h = W * sin(theta)
Using the formula:
T_h = 25.48 N * sin(7.5°) = 25.48 N * 0.121869 = 3.10 N
Step 4: Calculate the net force in the horizontal direction.
The net force (F_net) acting on the object in the horizontal direction is equal to the horizontal component of the tension:
F_net = T_h
Using the formula:
F_net = 3.10 N
Step 5: Use Newton's second law to find the acceleration of the car.
Newton's second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
F_net = m * a
where a is the acceleration.
Using the formula:
3.10 N = 2.6 kg * a
Rearranging the equation to solve for a:
a = 3.10 N / 2.6 kg ≈ 1.19 m/s²
Therefore, the acceleration of the car is approximately 1.19 m/s².