A simple pendulum has a bob of mass M. The bob is on a light string of length l. The string is fixed at C. At position A, the string if horizontal and the bob is at rest. The bob is released from A and swings to B, where the string is vertical. The tension in the string when the bob first reaches B is?
So to find tension I know that F(total) = Tension - (some other force), but I don't know for sure if that force is centripetal force or what.
To set up originally I chose Mg and know it is incorrect so I'm thinking that it is actually none of the above by using conservation of energy and knowing that centripetal force is mv^2/r where r in this case is l and v^2 is 2gh using conservation of energy where
mgh = 1/2mv^2
and then from there getting the formula. So the actual answer would be
m*2*g*h/l
does this sound correct?
Oh wait I think I figured it out from a video.
Total force is equal to mass time centripetal acceleration
Total-F(c) = m*a(c)
Where Total-F(c) = m* v^2/r
T - mg = m * v^2/r
where v^2 = 2gh and h = l and r = l
so you get
T-mg = m * (2*g*l)/l
from that you get
T-mg = 2mg
then finally
T = 3mg
I have no word about your helping me such a way
Thank you
No, your approach is not correct. The force balance at the bottom of the swing (position B) should involve the centripetal force and the weight of the bob.
At position B, the bob is at the lowest point of its swing, which means it has the maximum speed and is experiencing the maximum tension in the string. The tension in the string provides the centripetal force required to keep the bob moving in a circular path.
To find the tension at position B, you can set up the force balance equation:
Tension - Weight = Centripetal Force.
Weight is equal to the mass of the bob (M) multiplied by the acceleration due to gravity (g), so the equation becomes:
Tension - M * g = M * v^2 / l.
Here, v is the velocity of the bob at position B.
To find the velocity (v), you can use conservation of mechanical energy. The initial potential energy of the bob at position A is equal to its final kinetic energy at position B.
Using the conservation of energy equation:
M * g * l = 1/2 * M * v^2.
Simplifying this equation:
2 * g * l = v^2.
Now, substitute this value of v^2 into the force balance equation:
Tension - M * g = M * (2 * g * l) / l.
Simplifying further:
Tension - M * g = 2 * M * g.
Finally, solving for the tension:
Tension = 3 * M * g.
So, the tension in the string when the bob first reaches position B is 3 times the weight of the bob, which is 3 * M * g.
Your approach to solving the problem is correct, but the formula you mentioned is not accurate for finding the tension in the string when the bob first reaches position B. Let me explain the correct approach to finding the tension.
When the bob is at position A, the tension in the string is equal to the weight of the bob, which is its mass (M) multiplied by the acceleration due to gravity (g). So the tension at point A is TensionA = M * g.
As the bob swings to position B, it loses potential energy and gains kinetic energy. When it reaches position B, all its potential energy is converted into kinetic energy. So we can equate the potential energy at point A to the kinetic energy at point B:
M * g * l = 0.5 * M * v^2
Where v is the velocity of the bob at position B.
We can rearrange this equation to solve for the velocity v:
v = sqrt(2 * g * l)
Now, for the bob to reach position B, a centripetal force must act on it. This centripetal force is provided by the tension in the string at that point. The centripetal force is given by:
Fcentripetal = (M * v^2) / l
Plugging in the value of v we found earlier:
Fcentripetal = (M * (sqrt(2 * g * l))^2) / l
Simplifying this expression:
Fcentripetal = M * 2 * g
Therefore, the tension in the string when the bob reaches position B is equal to the centripetal force, which is:
TensionB = Fcentripetal = M * 2 * g
So, the correct answer for the tension in the string when the bob first reaches position B is TensionB = M * 2 * g.
I hope this clears up any confusion and helps you understand the correct approach to solving this problem.