A 3.25-kg object is attached to a vertical rod by two strings as shown in the figure below. The object rotates in a horizontal circle at constant speed 6.60 m/s.
(a) Find the tension in the upper string.
N
(b) Find the tension in the lower string.
N
Solution
To find the tensions in the upper and lower strings, we can use the concept of centripetal force. Centripetal force is the force that keeps an object moving in a circular path. In this case, the tension in the strings provides the centripetal force that keeps the object rotating in a horizontal circle.
Let's start by analyzing the forces acting on the object. There are two forces acting on the object: the force of gravity (mg) and the tension in the upper string (T1). The force of gravity acts vertically downward, and the tension in the upper string acts at an angle (θ) with respect to the vertical.
The tension in the upper string can be resolved into two components: T1sin(θ) acting upward and T1cos(θ) acting horizontally.
Since the object is rotating at a constant speed, the net force acting on it must be directed towards the center of the circle. This net force is provided by the horizontal component of the tension, T1cos(θ).
The centripetal force (Fc) is given by the equation:
Fc = m * v^2 / r
where m is the mass of the object, v is the velocity, and r is the radius of the circle.
In this case, the centripetal force is provided by the tension in the upper string:
Fc = T1cos(θ)
Setting the centripetal force equal to the tension in the upper string, we can solve for T1:
T1cos(θ) = m * v^2 / r
Now, let's calculate the values needed to find the tension in the upper string.
Given:
- Mass of the object (m) = 3.25 kg
- Constant speed (v) = 6.60 m/s
Assuming the length of the upper string is the radius of the circle, we can find the tension in the upper string by calculating the centripetal force.
First, let's find the centripetal force (Fc) using the equation:
Fc = m * v^2 / r
Substituting the given values:
Fc = (3.25 kg) * (6.60 m/s)^2 / r
Now, we need to find the angle (θ) in order to determine the horizontal component of the tension. The angle can be determined by trigonometry. If we assume that the upper string makes an angle α with the vertical, it follows that θ = 90° - α.
To find θ, we need to determine α using the given information. The object is in equilibrium, so the sum of the vertical forces must be zero:
T1sin(α) + mg = 0
Solving for sin(α):
sin(α) = -mg / T1
Now we can find θ:
θ = 90° - α
Finally, we can substitute the values of θ and Fc into the equation T1cos(θ) = Fc to find the tension in the upper string, T1.
For part (b), we can find the tension in the lower string by considering the vertical component of the tension, T2sin(θ), balancing the force of gravity (mg).