A toy train rolls around a horizontal 1.0-m-diameter track. The coefficient of rolling friction is 0.10. How long does it take the train to stop if it's released with an angular speed of 30 rpm?

Verified step by step guidance

1. Convert the given angular speed from revolutions per minute (rpm) to radians per second. Use the formula: \( \omega = \text{rpm} \times \frac{2\pi}{60} \), where \( \omega \) is the angular speed in radians per second.
2. Determine the radius of the circular track. Since the diameter is given as 1.0 m, the radius \( r \) is \( r = \frac{1.0}{2} = 0.5 \; \text{m} \).
3. Calculate the initial linear speed of the train using the relationship between angular speed and linear speed: \( v = \omega \cdot r \), where \( v \) is the linear speed, \( \omega \) is the angular speed, and \( r \) is the radius.
4. Determine the deceleration caused by rolling friction. The force of rolling friction is \( F_f = \mu \cdot m \cdot g \), where \( \mu \) is the coefficient of rolling friction, \( m \) is the mass of the train, and \( g \) is the acceleration due to gravity. The deceleration \( a \) is then \( a = \frac{F_f}{m} = \mu \cdot g \).
5. Use the kinematic equation \( v_f = v_i + a \cdot t \) to solve for the time \( t \) it takes for the train to stop. Here, \( v_f = 0 \) (final speed), \( v_i \) is the initial linear speed, and \( a \) is the deceleration. Rearrange the equation to find \( t = \frac{-v_i}{a} \).

Key Concepts

Rolling Friction

Rolling friction is the resistance that occurs when an object rolls over a surface. It is generally less than sliding friction and is influenced by factors such as the surface texture and the material of the rolling object. In this scenario, the coefficient of rolling friction (0.10) indicates how much force opposes the motion of the toy train as it rolls around the track.

Angular Speed

Angular speed refers to the rate at which an object rotates around an axis, measured in radians per second or revolutions per minute (rpm). In this question, the toy train is released with an angular speed of 30 rpm, which needs to be converted to a more usable unit for calculations. Understanding angular speed is crucial for determining how long it takes for the train to come to a stop due to friction.

Deceleration and Time to Stop

Deceleration is the rate at which an object slows down, often caused by frictional forces acting against its motion. To find the time it takes for the toy train to stop, one must calculate the deceleration due to rolling friction and then apply kinematic equations. This involves understanding the relationship between initial angular speed, deceleration, and time, allowing for the determination of how long the train will take to halt.