A pendulum is composed of a mass m suspended from the ceiling by a massless string of length L. If the mass is pulled back so that it is raised an amount L/4 and then released, what is the tension on the string at the bottom of the swing?
A.3 mg
B.4 mg
C.2 mg
D.5/2 mg
E.3/2 mg
F.1/2 mg
G.mg
To find the tension on the string at the bottom of the swing of the pendulum, we can use the principle of conservation of mechanical energy.
At the top point of the swing, when the mass is pulled back and released, it has potential energy equal to mgh, where h is the height raised.
Since the mass is raised an amount L/4, the height raised, h, is L/4.
Potential energy at the top = mgh = mg(L/4).
At the bottom point of the swing, all the potential energy is converted to kinetic energy. Kinetic energy = (1/2)mv^2.
Since the pendulum is at its lowest point, the velocity at the bottom is maximum. We can find the velocity at the bottom using the principle of conservation of mechanical energy.
Potential energy at the top = Kinetic energy at the bottom.
mg(L/4) = (1/2)mv^2.
We can cancel out the mass 'm' from both sides of the equation.
gL/4 = (1/2)v^2.
Multiplying both sides by 4/g, we get:
v^2 = 8gL.
Since the tension at the bottom is the centripetal force required to keep the mass moving in a circular path, we can use the relation:
Tension = mv^2/L.
Substituting the value of v^2 = 8gL, we get:
Tension = m(8gL)/L.
Canceling out the common factor 'm' and 'L', we are left with:
Tension = 8g.
The tension on the string at the bottom of the swing is 8 times the acceleration due to gravity 'g'. Therefore, the correct answer is G. mg.
To find the tension on the string at the bottom of the swing, we can use the principle of conservation of energy.
At the highest point of the swing, the mass has been raised an amount L/4 against gravity. The potential energy at this point is equal to mgh, where h is the height of the highest point.
Since the mass is raised L/4 above the lowest point, the height at the highest point is L - L/4 = 3L/4.
Therefore, the potential energy at the highest point is mgh = m * g * (3L/4).
At the lowest point of the swing, all the potential energy has been converted to kinetic energy.
Therefore, the kinetic energy at the lowest point is equal to the potential energy at the highest point. So, we have:
1/2 * m * v^2 = m * g * (3L/4),
where v is the velocity at the lowest point.
Simplifying the equation, we get:
v^2 = 2 * g * (3L/4).
Taking the square root of both sides, we get:
v = √(2 * g * (3L/4)).
Using the equation for velocity of a pendulum, v = √(g * L), we can equate the two and solve for L:
√(2 * g * (3L/4)) = √(g * L).
Squaring both sides and simplifying, we get:
2 * g * (3L/4) = g * L.
Simplifying further, we get:
6g * L/4 = g * L.
Cancelling out g and rearranging the equation, we get:
6L/4 = L.
Simplifying again, we find:
6L = 4L.
Dividing both sides by L, we get:
6 = 4.
Since this is not true, our original equation of v = √(g * L) is incorrect.
This means that there is no defined value for the tension on the string at the bottom of the swing. The tension changes continuously as the pendulum moves back and forth. The correct answer is therefore not listed among the options given.