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

Verified step by step guidance

1. Convert the rotational speed from revolutions per minute (rpm) to radians per second (rad/s) using the conversion factor: \(1 \text{ rpm} = \frac{2\pi \text{ rad}}{60 \text{ s}}\).
2. Calculate the angular velocity (\(\omega\)) using the formula: \(\omega = 2\pi \times \text{frequency}\), where frequency is the converted value from step 1.
3. Determine the radius of the circle in meters, as the string length is given in centimeters.
4. Use the formula for centripetal force (\(F_c\)) to find the tension in the string: \(F_c = m \times r \times \omega^2\), where \(m\) is the mass of the block, \(r\) is the radius of the circle, and \(\omega\) is the angular velocity.
5. The tension in the string is equal to the centripetal force calculated in step 4, as it is the force that keeps the block moving in a circular path.

Key Concepts

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. For an object in uniform circular motion, this force is provided by tension in the string, which counteracts the object's inertia trying to move it in a straight line. The formula for centripetal force is F_c = m * v^2 / r, where m is mass, v is tangential velocity, and r is the radius of the circular path.

Tangential Velocity

Tangential velocity is the linear speed of an object moving along a circular path, measured at any point along the circumference. It can be calculated from the rotational speed (in revolutions per minute) and the radius of the circle. In this case, the tangential velocity can be derived from the formula v = 2 * π * r * (rpm / 60), where r is the radius in meters and rpm is the revolutions per minute.

Tension in the String

The tension in the string is the force exerted along the string that keeps the block moving in a circular path. It acts as the centripetal force necessary to maintain circular motion. In this scenario, the tension can be calculated using the centripetal force equation, where the tension equals the mass of the block multiplied by the square of its tangential velocity divided by the radius of the circle.