A light string can support a stationary hanging load of 26.4 kg before breaking. An object of mass m = 3.19 kg attached to the string rotates on a frictionless, horizontal table in a circle of radius r = 0.795 m, and the other end of the string is held fixed as in the figure below. What range of speeds can the object have before the string breaks?
0 to what?? please help!
To determine the range of speeds the object can have before the string breaks, we need to consider the tension in the string. The tension force in the string is responsible for providing the necessary centripetal force to keep the object moving in a circular path.
Let's start by analyzing the forces acting on the object when it is rotating in a circle. The two main forces are the tension in the string (T) and the object's weight (mg). Since the table is frictionless, there is no additional horizontal force acting on the object.
The tension force, T, provides the centripetal force for circular motion and is given by the equation:
T = m * v^2 / r
where m is the mass of the object, v is the velocity of the object, and r is the radius of the circle.
To find the range of speeds before the string breaks, we need to calculate the maximum tension the string can withstand. Given that the string can support a stationary hanging load of 26.4 kg before breaking, this is the maximum tension it can handle.
So, we set T equal to the maximum tension:
T = 26.4 kg * 9.8 m/s^2
Now, we can substitute the expression for tension, T, into the equation for circular motion, and rearrange the equation to solve for v:
26.4 kg * 9.8 m/s^2 = (3.19 kg * v^2) / 0.795 m
Solving for v:
v = sqrt((26.4 kg * 9.8 m/s^2 * 0.795 m) / 3.19 kg)
Calculating the value gives us:
v ≈ 5.369 m/s
So, the maximum speed the object can have before the string breaks is approximately 5.369 m/s.
Therefore, the range of speeds before the string breaks is from 0 m/s (when the object is at rest) to approximately 5.369 m/s.