Period and Tension of a Pendulum

In summary: T)## is a solution to this equation. If it is, then the period is T. Otherwise, the period is twice the original value.In summary, The tension in a pendulum is maximum when the pendulum is at the angle θg where θ_g=θ_0.
  • #1
postfan
259
0

Homework Statement


A simple pendulum experiment is constructed from a point mass m attached to a pivot by a massless rod of length
L in a constant gravitational field. The rod is released from an angle θ0 < π/2 at rest and the period of motion is
found to be T0. Ignore air resistance and friction.
19. At what angle θg during the swing is the tension in the rod the greatest?
(A) The tension is the greatest at the point θ_g = θ_0.
(B) The tension is the greatest at the point θ_g = 0.
(C) The tension is the greatest at an angle θ_g with 0 < θ_g < θ_0.
(D) The tension is constant.
(E) None of the above is true for all values of θ0 with 0 < θ0 < π/2.
20. What is the maximum value of the tension in the rod?
(A) mg
(B) 2mg
(C) mLθ0_/T_0^2
(D) mg sin θ_0
(E) mg(3 − 2 cos θ_0)
21. The experiment is repeated with a new pendulum with a rod of length 4L, using the same angle θ0, and the period of motion is found to be T. Which of the following statements is correct?
(A) T = 2T_0 regardless of the value of θ_0.
(B) T > 2T_0 with T ≈ 2T_0 if θ_0 << 1.
(C) T < 2T_0 with T ≈ 2T_0 if θ_0 << 1.
(D) T > 2T_0 for some values of θ_0 and T < 2T_0 for other values of θ_0.
(E) T_0 and T are undefined because the motion is not periodic unless θ_0 << 1.

Homework Equations

The Attempt at a Solution


For 19, I just drew an FBD and realized that the tension in a pendulum is just mgcos(θ) which is maximized when θ=0 so the answer is B.
For 20, when θ=0 (where the maximum tension is) from the formula the tension is just mg which is (A).
For 21, I used the formula T=2pi*sqrt(L/g) so quadrupling the length should double the period making it (A).

Are my answers (and most importantly my justifications) right?
 
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  • #2
postfan said:
For 19, I just drew an FBD and realized that the tension in a pendulum is just mgcos(θ) which is maximized when θ=0 so the answer is B.
For 20, when θ=0 (where the maximum tension is) from the formula the tension is just mg which is (A).
For 21, I used the formula T=2pi*sqrt(L/g) so quadrupling the length should double the period making it (A).

Are my answers (and most importantly my justifications) right?

19 and 21 are correct.

For 20, you'll need to find the velocity of the bob at the lowest point. The tension provides the necessary centripetal force for the semi-circular motion, so consider that as well.

FC = mv2/r
 
  • #3
postfan said:
For 19, I just drew an FBD and realized that the tension in a pendulum is just mgcos(θ)
You're overlooking centripetal acceleration. The rod not only has to counter gravity in the direction of the rod shaft, it also needs to provide the centripetal acceleration needed to make the bob travel in an arc. It doesn't change the answer to 19, but
postfan said:
when θ=0 (where the maximum tension is) from the formula the tension is just mg
it does change the answer to 20.
postfan said:
For 21, I used the formula T=2pi*sqrt(L/g)
That formula is only valid for small oscillations. Here θ0 can be up to π/2, which is definitely not small.
 
  • #4
Ok so basically if T-mg=mv^/r then T=mg+mv^/r , but how do I find v^2/r in terms of g and a trig function?

Also how do you find the period with large oscillations?
 
  • #5
postfan said:
Ok so basically if T-mg=mv^/r then T=mg+mv^/r , but how do I find v^2/r in terms of g and a trig function?

At the initial position, the pendulum will possesses potential energy. This will all be converted to kinetic evergy at the lowermost position. Get the value of 'v' using this.
 
Last edited:
  • #6
postfan said:
Also how do you find the period with large oscillations?
In general, that's a hard problem. Fortunately, all you care about here is how varying the length affects the period, keeping the angle fixed.
Can you write down the general differential equation for a simple pendulum (not using any small angle approximation)?
 
  • #7
Ok for 20 I did mg(L-Lcos(theta))=.5mv^2 and got v^2=2gL(1-cos(theta)). Substituting that into my "T" equation I got T=mg(3-2cos(theta)). How is that ?
For 21
774603e9b85eacfc9c3ba8f14df1e0f6.png
 
  • #8
postfan said:
Ok for 20 I did mg(L-Lcos(theta))=.5mv^2 and got v^2=2gL(1-cos(theta)). Substituting that into my "T" equation I got T=mg(3-2cos(theta)). How is that ?

Yes! You got that right, mate..!
 
  • #9
Cool now what about 21?
 
  • #12
postfan said:
Yes.
Now here's the clever bit. We want to find out how changing L will change the period, without actually solving the equation.
Suppose ##\theta = f(t)## is a solution to this differential equation with boundary condition ##f(0) = \theta_0##. Investigate whether ##h(t) = f(At)##, for some constant A, might be a solution to ##\frac {d^2 h}{dt^2} + \frac gL \sin(h(t))## with the same boundary condition.
 

FAQ: Period and Tension of a Pendulum

What is a pendulum?

A pendulum is a weight suspended from a fixed point that can swing back and forth due to the force of gravity.

What is the period of a pendulum?

The period of a pendulum is the time it takes for one complete swing or oscillation, from its starting point, back to that same point. It is typically measured in seconds.

How is the period of a pendulum calculated?

The period of a pendulum can be calculated using the formula T = 2π√(L/g), where T is the period in seconds, L is the length of the pendulum in meters, and g is the acceleration due to gravity (9.8 m/s²).

What factors affect the period of a pendulum?

The period of a pendulum is affected by two main factors: the length of the pendulum and the acceleration due to gravity. The longer the length of the pendulum, the longer the period. The stronger the force of gravity, the shorter the period. Other factors that may affect the period include air resistance and the weight of the pendulum.

How does tension affect a pendulum's period?

The tension in a pendulum's string or rod does not affect its period, as long as the tension remains constant. However, if the tension changes during the swing, it can affect the period and cause the pendulum to swing at a different rate.

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