A ball at the end of a 180 -cm-long string swings as a...

A ball at the end of a 180 -cm-long string swings as a pendulum as shown in Fig. 6-5. The ball's speed is $400 \mathrm{~cm} / \mathrm{s}$ as it passes through its lowest position. ( $a$ ) To what height $h$ above this position will it rise before stopping? (b) What angle does the pendulum then make to the vertical? Neglect all forms of friction. (a) The pull of the string on the ball is always perpendicular to the ball's motion and therefore does no work on the ball. Consequently, the ball's total energy remains constant; it loses KE but gains an equal amount of $\mathrm{PE}_{\mathrm{G}}$. That is, Change in KE $+$ change in $\mathrm{PE}_{\mathrm{G}}=0$ $$ \frac{1}{2} m v_{f}^{2}-\frac{1}{2} m v_{i}^{2}+m g h=0 $$ Since $v_{f}=0$ and $v_{i}=4.00 \mathrm{~m} / \mathrm{s}$, we find $h=0.8155 \mathrm{~m}$ as the height to which the ball rises. (b) From Fig. 6-5, $$ \cos \theta=\frac{L-h}{L}=1-\frac{0.8155}{1.80} $$ which gives $\theta=56.8^{\circ}$.

A ball at the end of a 180 -cm-long string swings as a pendulum as shown in Fig. 6-5. The ball's speed is $400 \mathrm{~cm} / \mathrm{s}$ as it passes through its lowest position. ( $a$ ) To what height $h$ above this position will it rise before stopping? (b) What angle does the pendulum then make to the vertical? Neglect all forms of friction.
(a) The pull of the string on the ball is always perpendicular to the ball's motion and therefore does no work on the ball. Consequently, the ball's total energy remains constant; it loses KE but gains an equal amount of $\mathrm{PE}_{\mathrm{G}}$. That is,
Change in KE $+$ change in $\mathrm{PE}_{\mathrm{G}}=0$
$$
\frac{1}{2} m v_{f}^{2}-\frac{1}{2} m v_{i}^{2}+m g h=0
$$
Since $v_{f}=0$ and $v_{i}=4.00 \mathrm{~m} / \mathrm{s}$, we find $h=0.8155 \mathrm{~m}$ as the height to which the ball rises.
(b) From Fig. 6-5,
$$
\cos \theta=\frac{L-h}{L}=1-\frac{0.8155}{1.80}
$$
which gives $\theta=56.8^{\circ}$.

Key Concepts

Geometric Relationships in Pendulum Motion

Trigonometric relationships, such as using the cosine of the angle formed by the string and the vertical, are used to relate the vertical height gained by the pendulum to the angular displacement. This key concept bridges the gap between linear and angular measurements in pendulum dynamics.

Work Done by Forces

The concept of work in physics relates to the force applied along the direction of displacement. In the context of pendulum motion, the tension force in the string acts perpendicular to the direction of the ball's motion and therefore does no work, allowing mechanical energy to be conserved.

Conservation of Mechanical Energy

In an isolated system where non-conservative forces (like friction) are neglected, the total mechanical energy remains constant. This means that any loss in kinetic energy is exactly balanced by a gain in gravitational potential energy when an object ascends, and vice versa during descent.

Kinetic and Potential Energy

Kinetic energy, expressed as one-half the mass multiplied by the velocity squared, represents the energy of motion, while gravitational potential energy, given by the product of mass, gravitational acceleration, and height, represents the energy stored due to an object's position in a gravitational field. The conversion between these two forms is fundamental in analyzing the motion of pendulums.

Transcript

0:00 Hello.

00:01 In this problem we have a pendulum that consists of a string of length 1 .8 or 180 centimeters and a ball at the end.

00:11 The speed of the ball at this lowest point here is 400 centimeters per second.

00:19 And the question is here first to determine the maximum height that this ball will reach to.

00:25 So at some point, this ball will keep rising until it reaches a maximum point here.

00:34 So the question is to determine how high does this ball rises, and then to determine the angle theta that the string make with the horizontal, with the vertical at this point.

00:48 So let's start by solving a here.

00:51 We have that, given that the string itself is perpendicular to the ball's motion, so it does no work on the ball and thus here no work or no forces are doing work in that bowl.

01:05 Thus, its energy is conserved.

01:07 So first thing is to note that the energy is conserved in that process here.

01:15 And from the energy conservation, we can then determine the height or the maximum height that this ball reaches.

01:23 So because of the energy is conserved, so if we call this position one and this, position 2 so the energy at position 1 is equal to the energy at position 2 so the energy of position 1 is the kinetic energy of the ball at this position plus the potential energy at this position and that's equal to the kinetic energy at this second position plus the potential energy of this second position if we assume that our lowest point is our zero point.

02:05 So here, that's our zero.

02:07 That's our reference point.

02:09 So here, the height at 1 is equal to 0...

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