Question

A playground merry-go-round has radius ﻿2.40 m2.40	ext{ m}2.40 m﻿ and moment of inertia ﻿2100 kg m22100	ext{ kg m}^22100 kg m2﻿ about a vertical axle through its center, and it turns with negligible friction. A child applies an ﻿18.0 N18.0	ext{ N}18.0 N﻿ force tangentially to the edge of the merry-go-round for ﻿15.0 s15.0	ext{ s}15.0 s﻿. If the merry-go-round is initially at rest, how much work did the child do on the merry-go-round?

Accepted Answer

First, understand that work done on an object is calculated using the formula: $$ W = F \cdot d \cdot \cos(	heta) $$, where $$ F  is $$the force applied, $$ d  is $$the distance over which the force is applied, and $$ 	heta  is $$the angle between the force and the direction of motion. In this case, the force is tangential, so $$ 	heta = 0 $$ degrees, and $$ \cos(0) = 1 . $$Next, calculate the distance $$ d $$ that the edge of the merry-go-round travels. Since the force is applied tangentially, the distance can be found using the arc length formula: $$ d = r \cdot \Delta	heta $$, where $$ r  is $$the radius of the merry-go-round and $$ \Delta	heta  is $$the angular displacement in radians. To find $$ \Delta	heta $$, use the relationship between angular acceleration $$ \alpha $$, time $$ t $$, and angular displacement: $$ \Delta	heta = \frac{1}{2} \cdot \alpha \cdot t^2 . $$First, calculate $$ \alpha $$ using the formula $$ \alpha = \frac{	au}{I} $$, where $$ 	au  is $$the torque and $$ I  is $$the moment of inertia. Torque $$ 	au  is $$given by $$ 	au = F \cdot r . $$Substitute the values into the torque formula: $$ 	au = 18.0 	ext{ N} \cdot 2.40 	ext{ m} . $$Then, use $$ \alpha = \frac{	au}{2100 	ext{ kg} \cdot 	ext{m}^2}  to $$find the angular acceleration. Finally, substitute $$ \alpha $$ and $$ t = 15.0 	ext{ s} $$ into $$ \Delta	heta = \frac{1}{2} \cdot \alpha \cdot t^2  to $$find the angular displacement. Use this $$ \Delta	heta  to $$calculate $$ d = r \cdot \Delta	heta $$, and then substitute $$ F $$, $$ d $$, and $$ \cos(0) = 1 $$ into the work formula $$ W = F \cdot d \cdot \cos(	heta)  to $$find the work done by the child.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Moment of Inertia

Torque

Work-Energy Principle