Question 157: Motion in 2D & Vectors

Question

A bob is whirled in a horizontal circle by means of a string at an initial speed of $$10 \mathrm{~rpm}$$. If the tension in the string is quadrupled while keeping the radius constant, the new speed is:

Accepted Answer

To solve this problem, we need to understand the relationship between the tension in the string and the speed of the bob whirling in a horizontal circle. The centripetal force acting on the bob is provided by the tension in the string, and it can be given by the formula:
$$ F = \frac{m v^2}{r} $$
where:
F
is the centripetal force (or tension in the string)
m
is the mass of the bob
v
is the tangential speed of the bob
r
is the radius of the circle
According to the problem, the initial speed of the bob is $$10 \mathrm{~rpm}$$, and the radius is kept constant. LetÃ¢ÂÂs denote the initial tension in the string as $$T_1$$ and the new tension as $$T_2$$. Given that the tension in the string is quadrupled, we have:
$$ T_2 = 4 T_1 $$
Also, the centripetal force can be written in terms of tension:
$$ T_1 = \frac{m v_1^2}{r} $$
$$ T_2 = \frac{m v_2^2}{r} $$
By substituting $$T_2 = 4 T_1$$ into the second equation, we get:
$$ 4 T_1 = \frac{m v_2^2}{r} $$
We know from the first equation that:
$$ T_1 = \frac{m v_1^2}{r} $$
Substituting this into our equation for $$T_2$$:
$$ 4 \left(\frac{m v_1^2}{r}ight) = \frac{m v_2^2}{r} $$
The masses and radii cancel out, leaving us with:
$$ 4 v_1^2 = v_2^2 $$
Taking the square root of both sides:
$$ v_2 = 2 v_1 $$
The initial speed $$v_1$$ is given as $$10 \mathrm{~rpm}$$. Therefore, the new speed $$v_2$$ is:
$$ v_2 = 2 	imes 10 \mathrm{~rpm} = 20 \mathrm{~rpm} $$
Hence, the new speed of the bob is
20 rpm
. The correct answer is Option A.

Answer

20 rpm

Answer

40 rpm

Answer

5 rpm

Answer

10 rpm

Motion in 2D & Vectors

A bob is whirled in a horizontal circle by means of a string at an initial speed of $10 \mathrm{~rpm}$. If the tension in the string is quadrupled while keeping the radius constant, the new speed is:

Choose the correct answer: