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Chapter 10: Problem 48

As the temperature is increased, the period of pendulum, (A) Increases as its effective length increases even though its centre of mass still remains at the centre of the bob. (B) Decreases as its effective length increases even through its centre of mass still remains at the centre of the bob. (C) Increases as its effective length increases due to shifting to centre of mass below the centre of the bob. (D) Decreases as its effective length remains same but the centre of mass shifts above the centre of the bob.

Short Answer

Expert verified
The correct answer is Option C: The period of a pendulum increases as its effective length increases due to temperature increase. This is because the period is directly related to the effective length, as shown in the formula \(T = 2\pi \sqrt{\frac{l}{g}}\). The position of the centre of mass does not directly impact the period of the pendulum.

Step by step solution

01

Option A: Increases as its effective length increases even though its centre of mass still remains at the centre of the bob.

The period of a pendulum is given by the formula \(T = 2\pi \sqrt{\frac{l}{g}}\), where \(T\) is the period, \(l\) is the length of the pendulum, and \(g\) is the acceleration due to gravity. As temperature increases, the effective length of the pendulum may indeed increase due to thermal expansion, but the position of the centre of mass does not impact the period directly. Therefore, we cannot conclude if this option is correct or not without considering the other options as well.
02

Option B: Decreases as its effective length increases even through its centre of mass still remains at the centre of the bob.

This option contradicts the formula for the period of a pendulum. As we have stated earlier, the period of a pendulum is given by \(T = 2\pi \sqrt{\frac{l}{g}}\). If the effective length of the pendulum increases, the period should also increase, not decrease. Therefore, this option is incorrect.
03

Option C: Increases as its effective length increases due to shifting to the centre of mass below the centre of the bob.

This option correctly states that the period of a pendulum will increase as its effective length increases. However, the reason given (shifting of the centre of mass below the centre of the bob) is not the correct explanation for this behavior. As previously mentioned, the position of the centre of mass does not directly impact the period of the pendulum. However, this option still remains partially correct as it states that the period would increase as the effective length increases with temperature.
04

Option D: Decreases as its effective length remains same but the centre of mass shifts above the centre of the bob.

This option contradicts the formula for the period of a pendulum. The period of a pendulum is directly related to its effective length, as shown in the formula \(T = 2\pi \sqrt{\frac{l}{g}}\). If the effective length does not change, the period would also remain the same. Therefore, this option is incorrect. Based on our analysis of the given options, we can conclude that:
05

Answer

The correct answer is Option C, as it states that the period of a pendulum increases as its effective length increases due to temperature increase, despite the given reason (shifting of the centre of mass) being incorrect. The main aspect to consider in this case is the change in the effective length as a result of temperature increase.

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Most popular questions from this chapter

The radius of a metal sphere at room temperature \(T\) is \(R\) and the coefficient of linear expansion of the metal is \(\alpha .\) The sphere heated a little by a temperature \(\Delta T\) so that its new temperature is \(T+\Delta T\). The increase in the volume of the sphere is approximately. (A) \(2 \pi R \alpha \Delta T\) (B) \(\pi R^{2} \alpha \Delta T\) (C) \(4 \pi R^{3} \alpha \Delta T / 3\) (D) \(4 \pi R^{3} \alpha \Delta T\)

A long metallic bar is carrying heat from one end to the other under steady state. The variation of temperature \(\theta\) along the length \(x\) of the bar from its hot end is best described by which of the following.

Two liquids \(A\) and \(B\) are at \(32^{\circ} \mathrm{C}\) and \(24^{\circ} \mathrm{C}\). When mixed in equal masses, the temperature of the mixture is found to be \(28^{\circ} \mathrm{C}\). Their specific heats are in the ratio of (A) \(3: 2\) (B) \(2: 3\) (C) \(1: 1\) (D) \(4: 3\)

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Two liquids \(A\) and \(B\) are at \(32^{\circ} \mathrm{C}\) and \(24^{\circ} \mathrm{C}\). When mixed in equal masses, the temperature of the mixture is found to be \(28^{\circ} \mathrm{C}\). Their specific heats are in the ratio of (A) \(3: 2\) (B) \(2: 3\) (C) \(1: 1\) (D) \(4: 3\)
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