An 0.10 kg object is hanging from your rearview mirror. While you are accelerating 4.0 m/s2 the string makes an angle with the vertical. There is a weight force going down and tension force along the string. What is the tension force and the angle?
Let the tension force be T and the angle be A. Vertical and horizontal force balance equations tell you thatL
T cos A = m g
T sin A = m a
Divide the secnd equation by the first.
tan A = a/g
a = artan a/g
Now solve for T
Hint: Add the squares of each of the first two equations
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To find the tension force and the angle at which the string makes with the vertical, we can use Newton's second law and some trigonometry.
1. Start by calculating the net force acting on the object. This can be done using Newton's second law: F_net = m*a, where F_net is the net force, m is the mass of the object (0.10 kg), and a is the acceleration (4.0 m/s^2).
F_net = (0.10 kg) * (4.0 m/s^2) = 0.40 N
2. The net force acting on the object is equal to the vector sum of the weight force and the tension force. The weight force always acts vertically downwards and can be calculated using the formula: F_weight = m*g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_weight = (0.10 kg) * (9.8 m/s^2) = 0.98 N
3. To find the tension force, subtract the weight force from the net force: F_tension = F_net - F_weight.
F_tension = 0.40 N - 0.98 N = -0.58 N
The negative sign indicates that the tension force is directed in the opposite direction of the weight force.
4. Finally, to find the angle, we can use trigonometry. The angle between the string and the vertical can be determined using the relationship: tan(theta) = F_tension / F_weight.
tan(theta) = (-0.58 N) / (0.98 N) = -0.59
To find the angle, we can take the inverse tangent (tan^(-1)) of the value obtained.
theta = tan^(-1)(-0.59)
Using a calculator, we find theta ≈ -30.8 degrees.
Therefore, the tension force is approximately 0.58 N and the angle at which the string makes with the vertical is approximately -30.8 degrees.