If mass speed and radius of rotation of a body moving in a circular path are all increased by 50%, the necessary force required to maintain the body moving in the circular path will have to be increased by

The centripetal force required to keep a body of mass $m$ moving in a circular path of radius $r$ with a speed $v$ is given by the formula:

$F=\frac{m{v}^2}{r}$

According to the question, the mass, speed, and radius of rotation are all increased by 50%. This means the new values (denoted by prime symbols) can be expressed as:

New mass, $m^{\prime }=m+0.5m=1.5m$
New speed, $v^{\prime }=v+0.5v=1.5v$
New radius, $r^{\prime }=r+0.5r=1.5r$

Let us calculate the new force, $F^{\prime }$, using these new values:

$F^{\prime }=\frac{m^{\prime }{\left(v^{\prime }\right)}^2}{r^{\prime }}$

Substitute the expressions for the new values into the formula:

$F^{\prime }=\frac{\left(1.5m\right){\left(1.5v\right)}^2}{1.5r}$

Simplify the expression by expanding the squared term in the numerator:

$F^{\prime }=\frac{1.5m\cdot 2.25v^2}{1.5r}$

Cancel the common factor of 1.5 in both the numerator and the denominator:

$F^{\prime }=2.25\cdot \frac{m{v}^2}{r}$

Since the original force is $F=\frac{m{v}^2}{r}$, we can write:

$F^{\prime }=2.25F$

The fractional increase in the necessary force is calculated as:

$\frac{F^{\prime }-F}{F}=\frac{2.25F-F}{F}=1.25$

To find the percentage increase, multiply this fractional increase by 100:

$\text{Percentage Increase}=1.25\times 100=125\%$

Thus, the force must be increased by 125% to maintain the body moving in the circular path.