A $1.0 \mathrm{~kg}$ stone is attached to one end of a rope, which has negligible mass. The stone moves in a uniform circular motion in a vertical circle of radius $2.0 \mathrm{~m}$ . Calculate the tension at the bottom point, if the tension in the rope is zero at the topmost point.

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1.0 \mathrm{~kg}

stone is attached to one end of a rope, which has negligible mass. The stone moves in a uniform circular motion in a vertical circle of radius

2.0 \mathrm{~m}

. Calculate the tension at the bottom point, if the tension in the rope is zero at the topmost point.

Understand Tension at Bottom: To solve this problem, we need to understand that the tension in the rope at the bottom of the circle is the force that provides the centripetal force necessary for circular motion, in addition to supporting the weight of the stone due to gravity. The tension at the bottom is the sum of the centripetal force and the gravitational force acting on the stone.
Calculate Gravitational Force: First, let's calculate the gravitational force (weight) acting on the stone. The gravitational force is given by the formula $F_{\text{gravity}} = m \times g$ , where $m$ is the mass of the stone and $g$ is the acceleration due to gravity (approximately $9.8 \, \text{m/s}^2$ ).
Calculate Centripetal Force: Using the values given, we have $F_{\text{gravity}} = 1.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 9.8 \, \text{N}.$
Use Conservation of Energy: Next, we need to calculate the centripetal force required for the stone to move in a circle at the bottom point. The centripetal force is given by the formula $F_{\text{centripetal}} = m \cdot v^2 / r$ , where $m$ is the mass, $v$ is the velocity of the stone, and $r$ is the radius of the circle.
Calculate Total Tension: Since the tension at the topmost point is zero, all the centripetal force required for circular motion at the top must come from the gravitational force. Therefore, at the top, $F_{\text{gravity}} = F_{\text{centripetal\_top}}$ . This means that $m \cdot g = \frac{m \cdot v^2}{r}$ . We can solve for $v^2$ by multiplying both sides by $r$ and dividing by $m$ .
Calculate Total Tension: Since the tension at the topmost point is zero, all the centripetal force required for circular motion at the top must come from the gravitational force. Therefore, at the top, $F_{\text{gravity}} = F_{\text{centripetal\_top}}$ . This means that $m \cdot g = \frac{m \cdot v^2}{r}$ . We can solve for $v^2$ by multiplying both sides by $r$ and dividing by $m$ . $v^2 = g \cdot r$ . Plugging in the values, we get $v^2 = 9.8 \, \text{m/s}^2 \cdot 2.0 \, \text{m} = 19.6 \, \text{m}^2/\text{s}^2$ .
Calculate Total Tension: Since the tension at the topmost point is zero, all the centripetal force required for circular motion at the top must come from the gravitational force. Therefore, at the top, $F_{\text{gravity}} = F_{\text{centripetal\_top}}$ . This means that $m \cdot g = \frac{m \cdot v^2}{r}$ . We can solve for $v^2$ by multiplying both sides by $r$ and dividing by $m$ . $v^2 = g \cdot r$ . Plugging in the values, we get $v^2 = 9.8 \, \text{m/s}^2 \cdot 2.0 \, \text{m} = 19.6 \, \text{m}^2/\text{s}^2$ . Now we have the velocity squared at the topmost point, which is the same as the velocity squared at the bottom point because of conservation of energy in the absence of friction and air resistance. We can now calculate the centripetal force at the bottom using $F_{\text{centripetal\_bottom}} = m \cdot v^2 / r$ .
Calculate Total Tension: Since the tension at the topmost point is zero, all the centripetal force required for circular motion at the top must come from the gravitational force. Therefore, at the top, $F_{\text{gravity}} = F_{\text{centripetal\_top}}$ . This means that $m \cdot g = \frac{m \cdot v^2}{r}$ . We can solve for $v^2$ by multiplying both sides by $r$ and dividing by $m$ . $v^2 = g \cdot r$ . Plugging in the values, we get $v^2 = 9.8 \, \text{m/s}^2 \cdot 2.0 \, \text{m} = 19.6 \, \text{m}^2/\text{s}^2$ . Now we have the velocity squared at the topmost point, which is the same as the velocity squared at the bottom point because of conservation of energy in the absence of friction and air resistance. We can now calculate the centripetal force at the bottom using $F_{\text{centripetal\_bottom}} = m \cdot v^2 / r$ . $F_{\text{centripetal\_bottom}} = 1.0 \, \text{kg} \cdot 19.6 \, \text{m}^2/\text{s}^2 / 2.0 \, \text{m} = 9.8 \, \text{N}$ .
Calculate Total Tension: Since the tension at the topmost point is zero, all the centripetal force required for circular motion at the top must come from the gravitational force. Therefore, at the top, $F_{\text{gravity}} = F_{\text{centripetal\_top}}$ . This means that $m \cdot g = \frac{m \cdot v^2}{r}$ . We can solve for $v^2$ by multiplying both sides by $r$ and dividing by $m$ . $v^2 = g \cdot r$ . Plugging in the values, we get $v^2 = 9.8 \, \text{m/s}^2 \cdot 2.0 \, \text{m} = 19.6 \, \text{m}^2/\text{s}^2$ . Now we have the velocity squared at the topmost point, which is the same as the velocity squared at the bottom point because of conservation of energy in the absence of friction and air resistance. We can now calculate the centripetal force at the bottom using $F_{\text{centripetal\_bottom}} = m \cdot v^2 / r$ . $F_{\text{centripetal\_bottom}} = 1.0 \, \text{kg} \cdot 19.6 \, \text{m}^2/\text{s}^2 / 2.0 \, \text{m} = 9.8 \, \text{N}$ . Finally, we can find the total tension at the bottom by adding the gravitational force and the centripetal force. $T_{\text{bottom}} = F_{\text{gravity}} + F_{\text{centripetal\_bottom}}$ .
Calculate Total Tension: Since the tension at the topmost point is zero, all the centripetal force required for circular motion at the top must come from the gravitational force. Therefore, at the top, $F_{\text{gravity}} = F_{\text{centripetal\_top}}$ . This means that $m \cdot g = \frac{m \cdot v^2}{r}$ . We can solve for $v^2$ by multiplying both sides by $r$ and dividing by $m$ . $v^2 = g \cdot r$ . Plugging in the values, we get $v^2 = 9.8 \, \text{m/s}^2 \cdot 2.0 \, \text{m} = 19.6 \, \text{m}^2/\text{s}^2$ . Now we have the velocity squared at the topmost point, which is the same as the velocity squared at the bottom point because of conservation of energy in the absence of friction and air resistance. We can now calculate the centripetal force at the bottom using $F_{\text{centripetal\_bottom}} = m \cdot v^2 / r$ . $F_{\text{centripetal\_bottom}} = 1.0 \, \text{kg} \cdot 19.6 \, \text{m}^2/\text{s}^2 / 2.0 \, \text{m} = 9.8 \, \text{N}$ . Finally, we can find the total tension at the bottom by adding the gravitational force and the centripetal force. $T_{\text{bottom}} = F_{\text{gravity}} + F_{\text{centripetal\_bottom}}$ . $m \cdot g = \frac{m \cdot v^2}{r}$ $0$ .