A small bead of mass 100 g is able to slide along a frictionless wire.
The wire is bent into a semicircle of radius 10 cm and rotates about a
vertical axis with angular velocity 12.5 rad/s.
(i) Draw a free body diagram depicting the forces acting on the bead
(ii) For what angle θ will the bead remain stationary relative to the rotating
wire?
(i) Free body diagram:
In the diagram, we have the bead located at the top of the semicircle (point A on the wire). The forces acting on the bead are:
1. Gravitational force (mg): This force acts vertically downward at the center of mass of the bead.
2. Normal force (N): This force acts perpendicular to the wire and counterbalances the gravitational force.
3. Centripetal force (F_cp): This force acts radially inward, towards the center of the semicircle, and is responsible for keeping the bead in circular motion.
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N
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| m |
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F_cp
↑
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mg
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(ii) To determine the angle θ at which the bead will remain stationary relative to the rotating wire, we need to consider the equilibrium condition. In this case, the net force acting on the bead should be zero.
The forces acting on the bead are the gravitational force (mg) and the normal force (N). The centripetal force (F_cp) does not contribute to the equilibrium condition since it acts perpendicular to the direction of motion.
At the top of the semicircle, the net force is given by the difference between the gravitational force and the normal force:
Net force = mg - N
For the bead to remain stationary, the net force should be zero. Therefore, we have:
mg - N = 0
Since the bead is in contact with the wire, the normal force is equal in magnitude to the gravitational force. Thus, we can rewrite the equation as:
mg - mg = 0
This simplifies to:
0 = 0
Since the equation holds true for any value of θ, the bead will remain stationary relative to the rotating wire at all angles θ.
To answer this question, let's start by drawing the free body diagram depicting the forces acting on the bead.
(i) Free Body Diagram:
Since the bead is sliding along a frictionless wire, the only two forces acting on the bead are the gravitational force (mg) and the centripetal force (Fc). The gravitational force always acts vertically downward, while the centripetal force acts towards the center of the semicircle.
To draw the free body diagram, you can represent the bead as a dot and draw two arrows representing these forces. The gravitational force should be pointing vertically downward, and the centripetal force should be pointing towards the center of the semicircle.
(ii) Finding the Angle θ:
To find the angle θ at which the bead remains stationary relative to the rotating wire, we need to consider the balance of forces. When the bead is stationary, the centripetal force acting towards the center of the semicircle is equal to the gravitational force acting downward.
The centripetal force (Fc) is given by the formula:
Fc = m * v^2 / r,
where:
- m is the mass of the bead (0.1 kg),
- v is the tangential velocity of the bead, and
- r is the radius of the semicircle (0.1 m).
Since the bead is stationary, the tangential velocity (v) is zero. Therefore, the centripetal force (Fc) will also be zero. On the other hand, the gravitational force (mg) remains the same.
Equating the centripetal force to the gravitational force, we have:
mg = m * v^2 / r
Since v = 0, the equation simplifies to:
mg = 0
As a result, the bead remains stationary at any angle θ.