A 0.20-kg ball on a stick is whirled on a vertical circle at a constant speed. When the ball is at the three o'clock position, the stick tension is 16N. Find the tensions in the stick when the ball is at the twelve o'clock and at the six o'clock positions.
To find the tensions in the stick at the twelve o'clock and six o'clock positions, we'll need to consider the forces acting on the ball at those points.
At the three o'clock position, the only force acting on the ball is its weight (mg), which is directed downward. The tension in the stick is equal to the centripetal force required to keep the ball moving in a circle at a constant speed.
Let's find the centripetal force at the three o'clock position. The centripetal force is given by the formula: Fc = (mv^2) / r, where m is the mass of the ball, v is its velocity, and r is the radius of the circular path.
We're given that the mass of the ball is 0.20 kg. Since the ball is moving at a constant speed, its velocity remains the same throughout the circular path. Therefore, the centripetal force at the three o'clock position is equal to the tension in the stick, which is 16 N.
Fc = Tension at three o'clock position = 16 N
To find the tensions at the twelve o'clock and six o'clock positions, we need to consider the forces acting on the ball at those points. At the twelve o'clock position, the tension in the stick will be at its maximum, while at the six o'clock position, the tension will be at its minimum.
At the twelve o'clock position, the centripetal force is the sum of the tension and the weight of the ball. The tension acts downward, while the weight acts downward.
Let's assume the tension at the twelve o'clock position is T12. The centripetal force at the twelve o'clock position is given by the formula:
Fc = Tension at twelve o'clock position + Weight of the ball
Since the ball is at the highest point of the vertical circle, the centripetal force must equal the weight of the ball.
T12 + mg = mg
Simplifying the equation, we find:
T12 = mg - mg = 0 N
Therefore, the tension in the stick at the twelve o'clock position is 0 N.
At the six o'clock position, the centripetal force is the difference between the weight of the ball and the tension in the stick, with both forces acting downward. Let's assume the tension at the six o'clock position is T6. The centripetal force at the six o'clock position is given by the formula:
Fc = Weight of the ball - Tension at six o'clock position
Since the ball is at the lowest point of the vertical circle, the centripetal force must equal the weight of the ball.
mg = mg - T6
Simplifying the equation, we find:
T6 = mg - mg = 0 N
Therefore, the tension in the stick at the six o'clock position is also 0 N.
In summary, the tension in the stick is 16 N at the three o'clock position, and it is 0 N at both the twelve o'clock and six o'clock positions.
To find the tensions in the stick when the ball is at the twelve o'clock and six o'clock positions, we need to consider the forces acting on the ball at these positions.
1. At the three o'clock position, only two forces act on the ball: the tension force in the stick and the gravitational force (weight) acting downwards. The ball moves in a circle, so the net force acting inward must provide the centripetal force.
The weight of the ball can be calculated as:
Weight = mass * gravity
Weight = 0.20 kg * 9.8 m/s^2
Weight = 1.96 N
Since the ball is moving at a constant speed, the tension force in the stick must equal the centripetal force at the three o'clock position:
Tension + Weight = Centripetal force
16 N + 1.96 N = Centripetal force
Centripetal force = 17.96 N
2. At the twelve o'clock position, the only force acting on the ball is the tension force in the stick. The weight force and the centripetal force are both directed downwards, so they cancel each other out.
Therefore, at the twelve o'clock position, the tension in the stick is equal to the weight of the ball:
Tension = Weight
Tension = 1.96 N
3. At the six o'clock position, the only force acting on the ball is the tension force in the stick. The weight force and the centripetal force are both directed upwards, so they add up.
Therefore, at the six o'clock position, the tension in the stick is the sum of the weight of the ball and the centripetal force:
Tension = Weight + Centripetal force
Tension = 1.96 N + 17.96 N
Tension = 19.92 N
So, the tensions in the stick at the twelve o'clock and six o'clock positions are 1.96 N and 19.92 N, respectively.