The 0.5-kg ball is fired up the smooth vertical circular track using the spring plunger. The plu

The 0.5-kg ball is fired up the smooth vertical circular...

Key Concepts

Conservation of Mechanical Energy

This concept involves the idea that in the absence of non-conservative forces (like friction), the total mechanical energy of a system (the sum of kinetic and potential energies) remains constant. In problems where a spring is used to launch an object and the object moves against gravity along a track, the energy stored in the spring is converted into kinetic energy and gravitational potential energy as the object moves along the track.

Spring Potential Energy

Spring potential energy refers to the energy stored in a compressed or extended spring, which is given by (1/2) k x², where k is the spring constant and x is the compression or extension distance. This energy is released when the spring is allowed to return to its equilibrium position, acting as the source of the kinetic energy imparted to the object.

Circular Motion Dynamics and Centripetal Force

When an object moves along a curved path, such as a circular track, it requires a centripetal force directed toward the center of the circle to maintain its circular motion. This force is provided by the net inward force acting on the object, which can come from components of gravity, spring force, or normal force. The analysis of these forces is essential to determine the motion characteristics along the track.

Condition for Loss of Contact

The condition for loss of contact on a curved path occurs when the normal force, which is the force exerted by the surface on the object, becomes zero. At this point, the only force acting toward the center is a component of gravity (or other forces), and the object is on the verge of leaving the track. Establishing the criteria for when the normal force is zero is crucial for problems involving departure from a surface.

Transcript

00:01 So here in this question firstly let's try to imagine the situation here so we have a vertical circular track and we have the ball here let's assume that the ball is here and there is spring plunger over here it is compressed it is compressed 0 .08 meters now the motion of the ball will be like this and the forces acting on the ball will be let's assume that the ball is here and the force is acting on the ball will be the gravitational force which is mg the normal force which is n and one is centrifugal force which is acting towards the center of the circle is fc now the normal force is the force that the track exerts on the ball to prevent it from falling the centripetal force is the force that keeps the ball moving in circular path when the ball is bottom of the track when theta is zero degrees here theta is zero degrees because it is making zero degrees with the floor now the centripetal force is equal to the normal force in this condition the centripetal force is equals to the normal force however as the ball moves up on the track the centripetal force decreases and this is because the ball is moving slower and the radius of the curvature of the track is increasing at some point the centripetal force will become so small that no longer enough to keep the ball on the track this is when the ball leave the track now to determine the ball must be pulled back and released so that it will begin to leave the track when theta is 135 degrees so we can use the equation the centripetal force fc is equals to mg cos theta where fc is the centripetal force m is the mass of the ball g is acceleration due to gravity and theta is the angle between the vertical theta is the angle between vertical between vertical and line connecting the center of the track line connecting line connecting the center line connecting the center we can also use the equation of centripetal force that is fc is equals to mv square by r which is the formula for centripetal force where v is the velocity r is…

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