A small lead ball of mass 2kg is suspended at the end of a light string 1m in length. A small peg, 0.5m below the suspension point, catches the string in its swing. The ball is set swinging through small angles.

A) What is the period of the pendulum?

B) The ball has started swinging on the side that does not catch the peg, at an initial height 0.05m above the low point. How high does it rise on the side where the peg restricts the pendulum length to 0.5m?

Period now consists of two "half" periods. You know the length of both halves (T=2pi sqrt (L/g) ) Add both periods for each of the lengths, and divide by 2.

for the height, KE will be zero. Initial PE= final PE. Solve for final height from that.

Can you please detail?

To find the period of the pendulum, we can use the formula:

T = 2π√(L/g)

where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

A) The length of the pendulum is given as 1m (since it is the length of the string). The acceleration due to gravity, g, is approximately 9.8 m/s^2. Substituting these values into the formula:

T = 2π√(1/9.8)

Using a calculator, we get:

T ≈ 2π√(0.102)

T ≈ 2π(0.319)

T ≈ 2.006 seconds

So, the period of the pendulum is approximately 2.006 seconds.

B) To find how high the ball rises on the side where the peg restricts the pendulum length to 0.5m, we can use conservation of energy. At the initial height of 0.05m, the ball has gravitational potential energy. When it reaches the maximum height, this potential energy is converted into kinetic energy, and the maximum height can be found using the conservation of energy principle.

The difference in potential energy between the initial and maximum heights is:

ΔPE = mgh

where m is the mass of the ball, g is the acceleration due to gravity, and h is the height difference.

In this case, m = 2kg, g = 9.8 m/s^2, and h = 0.5m - 0.05m = 0.45m.

Substituting these values into the formula:

ΔPE = 2kg * 9.8 m/s^2 * 0.45m

ΔPE = 8.82 J

This energy is converted into kinetic energy at the maximum height:

ΔPE = KE

where KE is the kinetic energy.

The kinetic energy can be calculated using the formula:

KE = (1/2)mv^2

where m is the mass of the ball and v is the velocity at the maximum height.

Simplifying the equation:

8.82J = (1/2) * 2kg * v^2

8.82J = v^2

Taking the square root of both sides:

v ≈ √8.82

v ≈ 2.97 m/s

Now, we need to find the height at this velocity using conservation of energy:

KE = (1/2)mv^2 = (1/2) * 2kg * (2.97 m/s)^2

KE ≈ 8.82J

This potential energy at the maximum height is equal to the gravitational potential energy at the minimum height, which can be calculated as:

PE = mgh

8.82J = 2kg * 9.8 m/s^2 * h_max

Solving for h_max:

h_max ≈ 0.45m

So, the ball rises to a height of approximately 0.45m on the side where the peg restricts the pendulum length to 0.5m.

To find the period of the pendulum, we can use the formula:

T = 2π√(L/g)

Where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

A) Given that the length of the pendulum is 1m, and the acceleration due to gravity is approximately 9.8 m/s², we can substitute these values into the formula:

T = 2π√(1/9.8)
T = 2π/√9.8
T = 2π/3.13
T ≈ 2.01 s

Therefore, the period of the pendulum is approximately 2.01 s.

B) To find how high the ball rises on the side where the peg restricts the pendulum length to 0.5m, we can use the conservation of energy principle.

At the initial height, the ball has gravitational potential energy, which is converted into kinetic energy as it swings down to the lowest point. From there, the kinetic energy is converted back into potential energy as the ball swings back up.

The total mechanical energy (E) of the system is conserved, so we can equate the initial potential energy (mgh) to the final potential energy (mgH), where H is the final height.

mgh = mgH

Since the mass (m) of the ball cancels out, we can simplify the equation:

gh = H

Given that the initial height (h) is 0.05m and the length of the pendulum (L) is 0.5m, we can substitute these values into the equation:

9.8 * 0.05 = H
H ≈ 0.49m

Therefore, the ball will rise to approximately 0.49m on the side where the peg restricts the pendulum length to 0.5m.