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Conservation of Energy

The string of...

Question

The string of a pendulum is attached to a bob and is released from its initial rest position. A fixed peg is present in the path of motion of the pendulum. What are the velocities of the bob at positions $B$ and $C$ respectively? The distances are as shown in the figure.

3.16 m/s

and

7.07 m/s

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7.07 m/s

and

3.16 m/s

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5.34 m/s

and

10.78 m/s

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7.07 m/s

and

6.32 m/s

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Solution

The correct option is D $7.07 m/s$ and $6.32 m/s$
As the bob is released, it swings downwards, Hence its potential energy is converted completely into kinetic energy at point $B$ . Taking reference level at point $B$

$K E_{B} = P E_{A} \Rightarrow \frac{1}{2} m {v_{B}}_{B}^{2} = m g h \Rightarrow {v_{B}}_{B}^{2} = 2 g h \Rightarrow v_{B} = \sqrt{2 g h} = \sqrt{2 \times 10 \times 2.5} \Rightarrow v_{B} = 7.07 m / s$
Therefore velocity of the bob at point $B$ is $7.07 m/s$ .

From the point $B$ to $C$ , only the thread below the peg can swing. Therefore, applying conservation of energy at points $B$ and $C$

$K E_{C} + P E_{C} = K E_{B} = P E_{A} \frac{1}{2} m {v_{C}}_{C}^{2} + m g h_{C} = m g h \frac{1}{2} m {v_{C}}_{C}^{2} = m g (h - h_{C}) \Rightarrow v_{C} = \sqrt{2 g (h - h_{C})} \Rightarrow v_{C} = 6.32 m / s$

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Similar questions

Q. The string of a pendulum is attached to a bob and is released from its initial rest position. A fixed peg is present in the path of motion of the pendulum. What are the velocities of the bob at positions

B

and

C

respectively? The distances are as shown in the figure.

Q. A simple pendulum of length L with a bob of mass m is deflected from its rest position by an angle θ and released (figure 8-E16). The string hits a peg which is fixed at a distance x below the point of suspension and the bob starts going in a circle centred at the peg. (a) Assuming that initially the bob has a height less than the peg, show that the maximum height reached by the bob equals its initial height. (b) If the pendulum is released with

θ = 90 ° and x = L / 2

, find the maximum height reached by the bob above its lowest position before the string becomes slack. (c) Find the minimum value of x/L for which the bob goes in a complete circle about the peg when the pendulum is released from

θ = 90 °

.

Figure

Q. A simple pendulum of length

L

having a bob of mass

m

is deflected from its rest position by an angle

θ

and released (figure). The string hits a peg which is fixed at a distance

x

below the point of suspension and the bob starts going in a circle centred at the peg.
(a) Assuming that initially the bob has height less than the peg, show that the maximum height reached by the bob equals its initial height
(b) If the pendulum is released with

θ = 90^{\circ}

and

x = L / 2

find the maximum height reached by the bob above its lowest position before the string becomes slack
(c) Find the minimum value of

X / L

for which the bob goes in a complete circle about the peg when the pendulum is released from

θ = 90^{\circ}

Q. A bob is attached to one end of a string other end of which is fixed at peg

A

. The bob is taken to a position where string makes an angle of

30^{o}

with the horizontal. On the circular path of the bob in vertical plane there is a peg

B

at a symmetrical position with respect to the position of release as shown in the figure. If

V_{c}

and

V_{a}

be the minimum tangential velocity in clockwise and anticlockwise directions respectively, given to the bob in order to hit the peg

B

then ratio

V_{c} : V_{a}

will be equal to

A simple pendulum of length L having a bob of mass m is deflected from its rest position by an angle $θ$ and released (figure) The string hits a peg which is fixed at a distance x below the point of suspension and the bob starts going in a circle centred at the peg. If the pendulum is released with $θ = 90^{\circ}$ and x = $\frac{L}{2}$ find the maximum height reached by the bob above its lowest position before the string becomes slack.

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