To solve the problem, let's go through each part systematically.
Given that the stone makes \(20\) revolutions in \(4\) seconds, we can utilize this information to find the frequency, period, speed, and centripetal force.
a) What is the frequency of the motion?
Frequency (\(f\)) is defined as the number of revolutions per second.
\[ f = \frac{\text{Number of revolutions}}{\text{Time (in seconds)}} \] \[ f = \frac{20 \text{ revolutions}}{4 \text{ seconds}} = 5 \text{ Hz} \]
b) What is its period?
The period (\(T\)) is the time taken for one complete revolution. It is the reciprocal of frequency.
\[ T = \frac{1}{f} \] \[ T = \frac{1}{5 \text{ Hz}} = 0.2 \text{ seconds} \]
c) What is the speed of the stone if the length of the string that holds the stone is 0.8 m?
The speed (\(v\)) of the stone in uniform circular motion can be calculated using the formula:
\[ v = \frac{C}{T} \]
where \(C\) (circumference) is given by:
\[ C = 2 \pi r \]
Here, \(r\) is the radius, which is the length of the string \(= 0.8 \text{ m}\).
\[ C = 2 \pi (0.8 \text{ m}) = 1.6 \pi \text{ m} \]
Now, substituting \(C\) into the speed formula:
\[ v = \frac{1.6 \pi}{0.2} = 8 \pi \text{ m/s} \] \[ v \approx 25.13 \text{ m/s} \quad \text{(using } \pi \approx 3.14\text{)} \]
d) What is the centripetal force acting on the stone to keep it in uniform circular motion if its mass is 30 g?
First, we need to convert the mass from grams to kilograms:
\[ m = 30 \text{ g} = 0.03 \text{ kg} \]
The centripetal force (\(F_c\)) can be calculated using the formula:
\[ F_c = \frac{mv^2}{r} \]
We already found \(v \approx 25.13 \text{ m/s}\) and \(r = 0.8 \text{ m}\):
\[ F_c = \frac{0.03 \text{ kg} \cdot (25.13 \text{ m/s})^2}{0.8 \text{ m}} \] Calculating \( (25.13)^2 \):
\[ (25.13)^2 \approx 632.8169 \text{ m}^2/\text{s}^2 \] Now substituting this value into the equation for centripetal force:
\[ F_c = \frac{0.03 \cdot 632.8169}{0.8} \approx \frac{18.9845}{0.8} \approx 23.73 \text{ N} \]
Summary of Results:
a) Frequency: \(5 \text{ Hz}\)
b) Period: \(0.2 \text{ seconds}\)
c) Speed: \(\approx 25.13 \text{ m/s}\)
d) Centripetal force: \(\approx 23.73 \text{ N}\)
3 answers