A stone is tied at one end of a 5 m long string and whirled in a vertical circle. The minimum speed required to just cross the topmost position is
To determine the minimum speed required to just cross the topmost position, we need to consider the forces acting on the stone at this point. At the topmost position, the gravitational force is acting downward, while the tension in the string is acting upward.
To keep the stone moving in a circle, the tension in the string must be greater than the gravitational force. At the topmost position, this means that the tension must be equal to the gravitational force plus some extra force to provide the necessary centripetal acceleration.
The gravitational force on the stone is given by:
F_gravity = m * g
where m is the mass of the stone and g is the acceleration due to gravity.
The tension in the string is given by:
F_tension = m * (v^2 / r)
where v is the speed of the stone, and r is the radius of the circle (which is equal to the length of the string).
Setting these two forces equal, we have:
m * g = m * (v^2 / r) + m * g_extra
where g_extra is the extra force needed to provide the necessary centripetal acceleration.
Simplifying, we get:
v^2 = g * r + g_extra
To just cross the topmost position, the centripetal acceleration must be equal to the gravitational acceleration, so we have:
g_extra = g
Substituting this into the previous equation, we get:
v^2 = 2 * g * r
Taking the square root of both sides, we get:
v = sqrt(2 * g * r)
Substituting r = 5 m and g = 9.81 m/s^2, we get:
v = sqrt(2 * 9.81 m/s^2 * 5 m) = 9.9 m/s
Therefore, the minimum speed required to just cross the topmost position is approximately 9.9 m/s.
the point is for ZERO net string force at the top
weight = mg down = mv^2/r down
There is ZERO tension in the string at that top point
v^2 / r = g
I agree with the numbers, not the logic.
You are correct, my mistake. At the topmost point of the circle, the tension in the string is equal to zero, since the weight of the stone provides the only vertical force acting on it. Therefore, the necessary condition for the stone to just cross the topmost position is that the tension equals zero and the gravitational force is equal to the centripetal force, which is given by:
m * g = m * (v^2 / r)
where m is the mass of the stone, g is the acceleration due to gravity, v is the speed of the stone, and r is the radius of the circle (equal to the length of the string).
Solving for v, we get:
v = sqrt(g * r)
Substituting g = 9.81 m/s^2 and r = 5 m, we get:
v = sqrt(9.81 m/s^2 * 5 m) = 9.9 m/s
Therefore, the minimum speed required for the stone to just cross the topmost position is approximately 9.9 m/s. Thank you for pointing out the mistake in my initial approach.
9.81 * 5 = 45.5
sqrt (45.5 )= 6.75 not 9.9
I do not know where you got the sqrt 2
You are correct, I apologize for the confusion. The correct calculation is:
v = sqrt(g * r)
Substituting g = 9.81 m/s^2 and r = 5 m, we get:
v = sqrt(9.81 m/s^2 * 5 m) = 6.26 m/s (rounded to two decimal places)
Therefore, the minimum speed required for the stone to just cross the topmost position is approximately 6.26 m/s, not 9.9 m/s. Thank you for bringing this to my attention.
To find the minimum speed required to just cross the topmost position in a vertical circle, we can use the concept of centripetal force.
The centripetal force required to keep an object moving in a circle is given by the equation:
F = (m * v^2) / r
Where:
F is the centripetal force,
m is the mass of the object,
v is the velocity of the object,
r is the radius of the circular path.
In this case, the tension in the string provides the centripetal force needed to keep the stone moving in a circle.
At the topmost position, the tension in the string should be equal to the weight of the stone, otherwise, the stone would fall down.
So, we can equate the tension in the string to the weight of the stone:
Tension = Weight
T = m * g
Where:
T is the tension in the string,
m is the mass of the stone,
g is the acceleration due to gravity.
In this problem, the length of the string (5 m) represents the radius of the circular path (r).
At the topmost position, the tension in the string is the maximum and should be equal to the weight of the stone:
T = m * g
Now, we substitute the tension in the previous equation for centripetal force:
m * g = (m * v^2) / r
Since we want to find the minimum speed required, we want to find the lowest possible value for v.
To find this, we can solve the equation for v:
v^2 = (g * r)
v = sqrt(g * r)
Substituting the values of g (acceleration due to gravity, approximately 9.8 m/s^2) and r (5 m):
v = sqrt(9.8 * 5)
v ≈ sqrt(49) ≈ 7 m/s
Therefore, the minimum speed required to just cross the topmost position is approximately 7 m/s.
To find the minimum speed required to just cross the topmost position, we can start by analyzing the forces acting on the stone at that point.
At the topmost position of the vertical circle, the stone is momentarily at rest. This means that the net force acting on the stone at that point is zero.
The forces acting on the stone can be broken down into two components: the tension in the string and the weight of the stone.
1. Tension in the string: At the topmost position, the tension in the string acts towards the center of the circle. This tension provides the necessary centripetal force to keep the stone moving in a circular path.
2. Weight of the stone: The weight acts vertically downwards due to gravity.
Since the net force is zero at the topmost position, we can equate the tension in the string to the weight of the stone:
Tension = Weight
The weight of the stone can be calculated using the formula:
Weight = mass × gravitational acceleration
Let's assume the mass of the stone is 'm' and the gravitational acceleration is 'g'.
Now, we can calculate the minimum speed required to just cross the topmost position by considering the tension in the string as the centripetal force:
Tension = (mass × velocity^2) / radius
Where:
- mass = mass of the stone (m)
- velocity = speed of the stone
- radius = radius of the circular path (equal to the length of the string, i.e., 5 m)
Setting the tension equal to the weight, we have:
(mass × gravitational acceleration) = (mass × velocity^2) / radius
Now, we can solve for the minimum speed required (velocity) by rearranging the equation:
velocity^2 = (radius × gravitational acceleration)
velocity = √(radius × gravitational acceleration)
Plugging in the values, we get:
velocity = √(5 m × 9.8 m/s^2)
Simplifying further, we have:
velocity = √(49 m^2/s^2)
velocity = 7 m/s
Therefore, the minimum speed required to just cross the topmost position is 7 m/s.