Question

A$500g$ball moves in a vertical circle on a $102-$cm-long string. If the speed at the top is $4.0m/s$, then the speed at the bottom will be $7.5m/s$.

a. What is the gravitational force acting on the ball?

b. What is the tension in the string when the ball is at the top? c. What is the tension in the string when the ball is at the bottom?

Short answer

a). The gravitational force acting on the ball is $4.90N$.

b). When the ball is at the top the tension in the string is $2.94N$.

c). When the ball is at the bottom the tension in the string is$32.5N$.

Step-by-step solution

Given Information (Part a) 

The mass of the ball is $m=500\mathrm{g}$, we have to calculate the gravitational Force on the mass.

Explanation (Pat a)

The gravitational force acting on a ball is:

$F=mg$

Where,

$g$is the acceleration due to gravity$=9.8m/s^2$

$m$is the mass of the ball$=0.5$

Therefore,

$F=0.5\times 9.8$

$F=4.90N$

Final Answer (Part a)

The gravitational force acting on the ball is$4.90N$.

Given Information (Part b)

The velocity at the top is $v=4.0\mathrm{m}/\mathrm{s}$, and the length of the string is $l=102\mathrm{cm}=1.02\mathrm{m}$, we have to calculate the Tension of the string at this position.

Explanation (Part b)

The required tension in the string at the bottom position is:

$T=\frac{m{v}^2}{L}+mg$

Here,

$T=$Tension in the string

$v=$Velocity at the top

$m=$Mass of the ball

$L=$Length of the string

role="math" localid="1648314532264" $T=\frac{0.5\times {\left(4\right)}^2}{102\times {10}^{-2}}+0.5\times 9.8$

role="math" localid="1648314543571" $T=2.93N$

Final Answer (Part b)

The tension in the string is $2.94N$.

Given Information (Part c)

The velocity of the ball at the bottom of the circle is $v^{\text{'}}=7.5\mathrm{m}/\mathrm{s}$, we have to calculate the tension of the string at the bottom.

Explanation (Part c)

To find the tension of the string, when the ball is at the bottom:

$T=mg+\frac{m{v}^2}{r}$

role="math" localid="1648315247523" $T=0.5\times 9.8+\frac{0.5\times 7.5^2}{1.02}$

$T=32.5N$

Final Answer (Part c)

When the ball is at the top the tension in the string is $32.5N$