To determine the maximum speed at which the string can rotate without breaking, we need to analyze the forces acting on the mass in circular motion. The key here is to understand that the tension in the string must not exceed the maximum tension it can handle, which is determined by the weight it can support. Let's break this down step by step.
Understanding the Forces in Circular Motion
When an object moves in a circle, it experiences a centripetal force that keeps it moving along that circular path. This centripetal force is provided by the tension in the string. The formula for centripetal force (F_c) is given by:
F_c = (m * v²) / r
- m is the mass of the object (2 kg in this case).
- v is the speed of the object.
- r is the radius of the circular path (0.8 m).
Maximum Tension in the String
The string can support a mass of 25 kg before breaking. The maximum tension (T_max) can be calculated using the weight of this mass:
T_max = m * g
Where g is the acceleration due to gravity, approximately 9.81 m/s². Thus:
T_max = 25 kg * 9.81 m/s² = 245.25 N
Setting Up the Equation
Now, we set the maximum tension equal to the centripetal force required to keep the 2 kg mass moving in a circle:
T_max = (m * v²) / r
Substituting the known values:
245.25 N = (2 kg * v²) / 0.8 m
Solving for Speed
To find v, we rearrange the equation:
v² = (245.25 N * 0.8 m) / 2 kg
v² = 98.1 m²/s²
Now, taking the square root of both sides gives:
v = √(98.1 m²/s²) ≈ 9.9 m/s
Final Consideration
Since we are looking for the maximum speed before the string breaks, we round this value to a more practical number. The closest option provided in your question is:
(B) 10 m/s
This means that the string will not break as long as the speed of the mass does not exceed approximately 10 m/s. If the speed were to increase beyond this, the tension would surpass the maximum limit, leading to the string breaking. Always remember to consider the forces at play in circular motion and the limits of the materials involved!
