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When an object moves in a circular path, it experiences a force directed towards the center of the circle, known as centripetal force. This force keeps the object on its curved trajectory by continually pulling it inwards. In our merry-go-round example, which involves rotational motion, the centripetal force is calculated using the formula:\[ F_{c} = m \cdot r \cdot \dot{\theta}^2 \]where:- \( m \) is the mass of the rotating object (the girl) at 50 kg,- \( r \) is the radius of the circular path at 4 m,- \( \dot{\theta} \) is the angular velocity at 1.5 rad/s. Plug the values into the formula to find the centripetal force:\[ F_{c} = 50 \cdot 4 \cdot (1.5)^2 = 450 \text{ N} \]This force is crucial for maintaining the circular motion of objects. It doesn't act as a separate force but is the resultant of various physical forces experienced, mainly due to the tension or normal force from the path or surface.

Gravitational force is a fundamental force acting on every object with mass. It pulls objects towards the center of the Earth. For any object sitting or moving around Earth's surface, this force is always acting downwards, towards the ground. To calculate the gravitational force acting on an object, use the following equation:\[ F_{g} = m \cdot g \]where:- \( m \) represents the object's mass, - \( g \) denotes the acceleration due to gravity, roughly 9.81 m/s².For our exercise, where the girl has a mass of 50 kg, calculate:\[ F_{g} = 50 \cdot 9.81 = 490.5 \text{ N} \]This is the force exerted by the earth on the girl, pulling her downward toward the ground.

Rotational motion refers to the movement of an object around a center or an axis. Unlike linear motion, rotational motion involves movements along a circular path, so the angular position of an object is key. In a merry-go-round scenario like the one described, the entire system of the girl on the horse undergoes rotational motion as it moves about the central pivot of the ride. This motion is governed by parameters such as: - Angular velocity, which is the rate of change of angular displacement. - Radius, which is the distance from the center to the point of interest (where the girl sits). A key component of rotational motion is that every point on the rotating object follows a curved path, and thus it constantly changes direction, inducing centripetal acceleration, which is directly linked to centripetal force.

Angular velocity is a measure of the rate at which an object rotates around an axis. It's a crucial concept in rotational dynamics as it helps us understand how fast something is spinning and how often it completes a rotation.It is commonly denoted by \( \dot{\theta} \) and is measured in radians per second (rad/s), a unit that quantifies angles in terms of pi (\( \pi \)). In our exercise involving the merry-go-round, the given angular velocity is 1.5 rad/s.Angular velocity establishes a relationship with both centripetal and tangential aspects of rotational motion. It determines how frequently the object retraces its path and directly impacts centripetal force as in the formula:\[ F_{c} = m \cdot r \cdot \dot{\theta}^2 \]In the context of the merry-go-round, angular velocity remains constant, simplifying our calculations and allowing us to understand how it affects other dynamic variables involved in the rotational motion.