Question

A boy uses a string to swing a 2.2 kg stone around his head. The string is 1.5 m long, and can bear a maximum tension of 1012 N. What is the maximum speed at which the boy can swing the stone without breaking the string?

a. 26.3 m/s
b. 15.78 m/s
c. 42.08 m/s

Answer 1

The maximum speed at which a boy can swing a 2.2 kg stone on a 1.5 m string without breaking the string, which can bear a maximum tension of 1012 N, is approximately 26.3 m/s.

Step-by-step explanation:

To solve this problem, we will use the concept of centripetal force. When an object is swung in a circular path, the force that keeps it moving in a circle is called the centripetal force. This force is provided by the tension in the string. The formula for centripetal force is:

F_c = frac{m cdot v^2}{r}

Where:
( F_c ) is the centripetal force,
( m ) is the mass of the object (the stone),
( v ) is the tangential (linear) speed of the object,
( r ) is the radius of the circular path (the length of the string).

We are given:
- the mass ( m = 2.2 ) kg,
- the maximum tension ( T_{max} = 1012 ) N, which is the maximum centripetal force that the string can provide before breaking,
- the radius ( r = 1.5 ) m.
To find the maximum speed ( v ), we can set the maximum tension developed in the string equal to the centripetal force and solve for ( v ):

T_{max} = F_c ]
1012 = \frac{2.2 cdot v^2}{1.5}

Now let's solve this equation for ( v ):

v^2 = frac{1012 cdot 1.5}{2.2}
v^2 = frac{1518}{2.2}
v^2 \approx 690 (rounded to three significant figures without a calculator)

Taking the square root of both sides to solve for ( v ):

approx sqrt690
approx 26.258

Since we are looking for an answer that matches the provided options, ( v approx 26.3 ) m/s matches one of them, so the correct answer to the problem is:

a. 26.3 m/s