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

Verified step by step guidance

1. Convert the mass of the block from grams to kilograms. Since 1 g = 0.001 kg, the mass \( m \) is \( 200 \times 0.001 = 0.2 \, \text{kg} \).
2. Convert the length of the string from centimeters to meters. Since 1 cm = 0.01 m, the radius \( r \) is \( 50 \times 0.01 = 0.5 \, \text{m} \).
3. Convert the rotational speed from revolutions per minute (rpm) to angular velocity \( \omega \) in radians per second. Use the formula \( \omega = \frac{2 \pi \times \text{rpm}}{60} \). Substituting \( \text{rpm} = 75 \), calculate \( \omega \).
4. Determine the centripetal force \( F_c \) acting on the block using the formula \( F_c = m r \omega^2 \). Substitute the values of \( m \), \( r \), and \( \omega \) into the equation.
5. Recognize that the tension in the string is equal to the centripetal force \( F_c \) because the string provides the force to keep the block moving in a circle. Thus, the tension in the string is \( T = F_c \).

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. It is essential for understanding the dynamics of circular motion, as it ensures that the object does not move off in a straight line. The formula for centripetal force (Fc) is Fc = mv²/r, where m is mass, v is velocity, and r is the radius of the circular path.

Tension in a String

Tension is the force transmitted through a string, rope, or cable when it is pulled tight by forces acting from opposite ends. In the context of circular motion, the tension in the string provides the necessary centripetal force to keep the block moving in a circle. The tension can be calculated by equating it to the centripetal force required for the block's circular motion.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves around a central point, expressed in radians per second or revolutions per minute (rpm). In this scenario, the block's angular velocity is crucial for determining its linear velocity, which is needed to calculate the centripetal force. The relationship between linear velocity (v) and angular velocity (ω) is given by v = ωr, where r is the radius of the circular path.