# String tension with an attached whirling mass

• MZA
In summary, the question asks for the maximum speed that a 0.40 kg mass can have while attached to a 0.75 m string without breaking the string, given that the maximum tension the string can withstand is 450 N. To solve this, we need to use the equations for force, centripetal acceleration, and velocity. The missing equation for this problem is a_{centripetal}=\frac{v^{2}}{r}.
MZA

## Homework Statement

A 0.40 kg mass, attached to the end of a .75 m string, is whirled around in a circular horizontal path. If the maximum tension that the string can withstand is 450 N, then what maximum speed can the mass have if the string is not to break?

F=m*a
T=r*F*sin(theta)
v=r*omega

## The Attempt at a Solution

I have absolutely no idea where to even begin with this. My idea was to find the acceleration when the force is 450 N, but that yields the linear acceleration not the angular one which does nothing for me. I am stumped.

Well even with the object being swung around at a constant (angular) speed you will still have a force. Because the direction of the velocity vector is changing, and acceleration is change in velocity. So you are looking for the acceleration inward, which is the centripetal acceleration, the force vector would be pointing from the object toward the center of the circle as it spins.

So the equations you are missing are:
$$a_{centripetal}=\frac{v^{2}}{r}$$
OR equivalently:
$$a_{centripetal}=\omega^{2}r$$

This is a classic example of circular motion, where the centripetal force (provided by the tension in the string) keeps the mass moving in a circular path. To find the maximum speed that the mass can have without breaking the string, we can use the formula for centripetal force:

F = m*v^2 / r

Where F is the force (in this case, the maximum tension of 450 N), m is the mass (0.40 kg), v is the velocity, and r is the radius of the circular path (0.75 m).

Rearranging the equation, we get:

v = √(F*r/m)

Substituting in the known values, we get:

v = √(450 N * 0.75 m / 0.40 kg) = 15.6 m/s

Therefore, the maximum speed that the mass can have without breaking the string is 15.6 m/s. Any faster and the tension in the string would exceed its maximum capacity and it would break.

## 1. What is string tension with an attached whirling mass?

String tension with an attached whirling mass is a scientific concept that refers to the force exerted on a string when it is attached to an object that is moving in a circular motion. This force is caused by the centripetal force needed to keep the object moving in a curved path.

## 2. How is string tension affected by the whirling mass?

The string tension is directly affected by the whirling mass because the greater the mass or speed of the object, the greater the centripetal force needed to keep it moving in a circular motion. This results in a higher string tension compared to a smaller or slower whirling mass.

## 3. What factors affect string tension with an attached whirling mass?

Apart from the mass and speed of the whirling object, the length of the string and the angle at which it is attached to the object can also affect the string tension. A longer string or a steeper angle will result in a higher string tension, while a shorter string or a shallower angle will result in a lower string tension.

## 4. How can string tension with an attached whirling mass be calculated?

The formula for calculating string tension with an attached whirling mass is T=mv^2/r, where T is the string tension, m is the mass of the object, v is its velocity, and r is the radius of the circular path. This formula is derived from the centripetal force equation, F=m(v^2)/r, where F is the centripetal force.

## 5. What are the practical applications of understanding string tension with an attached whirling mass?

Understanding string tension with an attached whirling mass is important in a variety of fields, such as engineering, physics, and sports. It can help engineers design structures that can withstand the force of a moving object, and it is also crucial in understanding the mechanics of circular motion in sports like gymnastics and figure skating.

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