A 3.75 kg object tied to a string is swung around a person’s head in a horizontal circle. The length of the string is 1.0 m and the maximum force the string can withstand without breaking is 25.0 N. What is the maximum speed the object may be swung without breaking the string?

To find the maximum speed at which the object can be swung without breaking the string, we can use the concept of centripetal force. The tension in the string provides the centripetal force required to keep the object moving in a circular path.

The centripetal force \( F_c \) required for an object moving in a circle is given by the formula:

\[
F_c = \frac{m v^2}{r}
\]

Where:
- \( m \) is the mass of the object (3.75 kg).
- \( v \) is the speed of the object.
- \( r \) is the radius of the circle, which in this case is the length of the string (1.0 m).

Given that the maximum force the string can withstand is 25.0 N, we can set the centripetal force equal to this maximum tension:

\[
F_c = 25.0 \, \text{N}
\]

Substituting into the centripetal force formula, we get:

\[
25.0 = \frac{3.75 v^2}{1.0}
\]

Now, we can solve for \( v^2 \):

\[
25.0 = 3.75 v^2
\]
\[
v^2 = \frac{25.0}{3.75}
\]
\[
v^2 = 6.6667
\]

To find \( v \), we take the square root:

\[
v = \sqrt{6.6667} \approx 2.58 \, \text{m/s}
\]

Thus, the maximum speed the object may be swung without breaking the string is approximately:

\[
\boxed{2.58 \, \text{m/s}}
\]