A 200 g block on a 50-cm-long string swings in a circle on a horizontal, frictionless table at 75 rpm. (a) What is the speed of the block?

Verified step by step guidance

1. Convert the rotational speed from revolutions per minute (rpm) to radians per second. Use the conversion factor where 1 rpm equals \(\frac{2\pi}{60}\) radians per second.
2. Calculate the angular velocity (\(\omega\)) using the formula \(\omega = 2\pi \times \text{frequency}\), where the frequency is the number of revolutions per second.
3. Determine the radius of the circle in meters. Since the string length is given in centimeters, convert it to meters by dividing by 100.
4. Use the relationship between linear speed (v), angular velocity (\(\omega\)), and radius (r) of the circle, which is given by \(v = r \times \omega\).
5. Substitute the values of \(\omega\) and radius into the formula to find the linear speed of the block.

Key Concepts

Centripetal Force

Centripetal force is the net force acting on an object moving in a circular path, directed towards the center of the circle. It is essential for maintaining circular motion and is provided by tension in the string in this scenario. The formula for centripetal force is F_c = m*v^2/r, where m is mass, v is speed, and r is the radius of the circular path.

Linear Speed

Linear speed refers to the distance traveled per unit of time. In circular motion, it can be calculated using the formula v = 2πr/T, where r is the radius and T is the period of rotation. In this case, the speed of the block can be derived from its rotational speed (in rpm) and the radius of the circular path.

Conversion of Units

Conversion of units is crucial in physics to ensure that all measurements are compatible. In this problem, the mass is given in grams, the length in centimeters, and the speed in revolutions per minute (rpm). To find the speed in meters per second, one must convert grams to kilograms, centimeters to meters, and rpm to seconds, ensuring accurate calculations.