Figure 8-31 shows a ball with mass m = 1.0 kg attached to the end of a thin rod with length L = 0.56 m and negligible mass. The other end of the rod is pivoted so that the ball can move in a vertical circle. (a) What initial speed must be given the ball so that it reaches the vertically upward position with zero speed? What then is its speed at (b) the lowest point and (c) the point on the right at which the ball is level with the initial point? (d) If the ball's mass were doubled, what would the answer to (a) be?

a thin rod, of length L = 1.6 m and negligible mass, that can pivot about one end to rotate in a vertical circle. A heavy ball of mass m = 9.2 kg is attached to the other end. The rod is pulled aside to angle è0 = 20° and released with initial velocity 0 = 0. What is the speed of the ball at the lowest point?

m=1kg

L=.56m

change in KE + change in PE = E

.5mv^2f + mghf = .5mv^2i + mghi

Even though this problem is a circle, the height variable in the potential energy equation shouldn't differ.

a)Use the bottom of the circle for h=0
mghi is canceled.
work = change in KE = .5mv^2f -.5mv^2i
You need the velocity

There is no KE initially so that equals 0. You're left with:

.5mv^2f=mghi solve for vf
(masses cancel out)

then plug v back into your intitial equation:
.5mv^2f-.5mv^2i= W

b) is just the negative of a

c) 0J because there is no KE and the change in PE = 0

d).5mv^2f=mghi solve for vf
because the masses aren't a factor in the final velocity, you can just go straight to the bottom equation

.5(2)mv^2f=mv^2f=W

To solve this problem, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of a system remains constant if there are no external forces acting on it.

Let's start by analyzing the situation at the highest point of the ball's motion, where it reaches the vertically upward position with zero speed.

(a) Initial speed at the highest point:
The ball must have enough initial speed to reach the highest point with zero speed. At the highest point, the only form of energy is potential energy (PE), since the ball's kinetic energy (KE) is zero.

Using the conservation of mechanical energy, we can equate the initial kinetic energy to the final potential energy:
KE_initial = PE_final

The initial kinetic energy can be calculated as:
KE_initial = (1/2) * m * v_initial^2

The final potential energy can be calculated as:
PE_final = m * g * h

Where:
m = mass of the ball (1.0 kg)
v_initial = initial velocity of the ball
g = acceleration due to gravity (9.8 m/s^2)
h = height of the highest point (which is equal to 2L since the ball moves in a vertical circle)

Setting these two equations equal to each other, we get:
(1/2) * m * v_initial^2 = m * g * (2L)

Simplifying and solving for v_initial:
v_initial^2 = 4 * g * L

Now substituting the given values:
v_initial^2 = 4 * 9.8 * 0.56
v_initial^2 = 21.9
v_initial ≈ 4.68 m/s

So, the initial speed required for the ball to reach the vertically upward position with zero speed is approximately 4.68 m/s.

(b) Speed at the lowest point:
At the lowest point, the energy is all converted into kinetic energy. Therefore, the speed at the lowest point can be calculated using the conservation of mechanical energy equation.

KE_initial + PE_initial = KE_final + PE_final

Since the initial and final potential energy remain the same, we can simplify the equation as:
KE_initial = KE_final

Substituting the values:
(1/2) * m * v_initial^2 = (1/2) * m * v_lowest^2

Simplifying and solving for v_lowest:
v_lowest^2 = v_initial^2

Substituting the given value of v_initial:
v_lowest^2 = (4.68)^2
v_lowest ≈ 21.9 m/s

So, the speed at the lowest point is approximately 21.9 m/s.

(c) Speed at the point on the right at which the ball is level with the initial point:
At this point, the potential energy is zero, and all the energy is in the form of kinetic energy. So, we can apply the conservation of mechanical energy equation again.

KE_initial + PE_initial = KE_final + PE_final

Again, since the initial and final potential energy remain the same, we can simplify the equation as:
KE_initial = KE_final

Substituting the values:
(1/2) * m * v_initial^2 = (1/2) * m * v_right^2

Simplifying and solving for v_right:
v_right^2 = v_initial^2

Substituting the given value of v_initial:
v_right^2 = (4.68)^2
v_right ≈ 21.9 m/s

So, the speed at the point on the right at which the ball is level with the initial point is also approximately 21.9 m/s.

(d) If the ball's mass were doubled:
If the mass of the ball were doubled, the initial speed required to reach the vertically upward position with zero speed would not change. It would still be approximately 4.68 m/s.

This is because the calculation for the initial speed does not depend on the mass of the ball.

However, the speeds at the lowest point and the point on the right at which the ball is level with the initial point would change. The speeds would also double because they depend on the square root of the mass. So, the speed at both these points would be approximately 43.8 m/s when the mass is doubled.

To solve this problem, you need to apply the principles of conservation of energy and centrifugal force.

(a) To find the initial speed that the ball must have in order to reach the vertically upward position with zero speed, we can start by analyzing the energy of the system.

At the highest point of the circle, the ball is at its maximum vertical position, meaning its kinetic energy is zero. Therefore, the initial speed must be such that all of the initial energy is in the form of gravitational potential energy.

The gravitational potential energy at the highest point is given by:
PE = mgh

Since the ball reaches the vertically upward position, the height h is equal to the length of the rod L. Thus:
PE = mgh = mLg

The initial energy is also equal to the sum of kinetic energy and gravitational potential energy at the highest point:
KE + PE = 0 + mLg

With no initial kinetic energy, the entire energy of the system is in the form of potential energy. Therefore:
KE_initial = 0

Using the conservation of energy, we can equate the initial energy to the final energy (at the lowest point of the circle):
KE_initial + PE_initial = KE_final + PE_final

Substituting the known values:
0 + mLg = KE_final + 0

Simplifying the equation, we find:
mLg = KE_final

(b) To find the speed at the lowest point of the circle, we need to consider the conservation of energy again. At the lowest point, all of the initial gravitational potential energy is converted into kinetic energy. Therefore:
mLg = (1/2)mv²

Simplifying the equation and canceling out the mass:
Lg = (1/2)v²

Solving for v, we find:
v = √(2gL)

Substituting the given values:
v = √(2 * 9.8 m/s^2 * 0.56 m)

Calculating the value:
v ≈ 4.44 m/s

So, the speed at the lowest point is approximately 4.44 m/s.

(c) To find the speed at the point on the right of the circle where the ball is level with the initial point, we can analyze the situation. At this point, the ball is at the same height as the initial point, so its potential energy is the same. However, its kinetic energy will be different.

Using the conservation of energy:
mLg = (1/2)mv²

Simplifying the equation and canceling out the mass:
Lg = (1/2)v²

Solving for v, we get:
v = √(2gL)

Substituting the given values:
v = √(2 * 9.8 m/s^2 * 0.56 m)

Calculating the value:
v ≈ 4.44 m/s

Hence, the speed at the point on the right where the ball is level with the initial point is also approximately 4.44 m/s.

(d) If the ball's mass were doubled, the answer to part (a) would change. Considering the equations we used previously, the mass of the ball appears in the expressions for both gravitational potential energy and kinetic energy.

In part (a), the gravitational potential energy is given by:
PE = mgh

If we double the mass, the potential energy would be doubled. Therefore, to find the new initial speed, we would use the equation:
2mLg = (1/2)mv²

Simplifying the equation and canceling out the mass:
Lg = (1/2)v²

Solving for v, we obtain:
v = √(2gL)

Substituting the given values:
v = √(2 * 9.8 m/s^2 * 0.56 m)

Calculating the value:
v ≈ 6.23 m/s

So, if the ball's mass were doubled, the new initial speed required to reach the vertically upward position with zero speed would be approximately 6.23 m/s.