¬Calculate the period of oscillation of a simple pendulum of length 1.8 m, with a bob of mass 2.2 kg. What assumption is made in this calculation?

(g = 9.8 m s - 2)
If the bob of this pendulum is pulled aside a horizontal distance of 20cm and
released, what will be the values of (i) the kinetic energy and (ii) the velocity of the bob at the lowest point of the swing?

T^2 = 4*pi^2(L/g)

T^2 = 39.48(1.8/9.8) = 7.25
T = 2.69 s.

Well, to calculate the period, 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. Plugging in the values we get, T = 2π√(1.8/9.8), which comes out to be around 1.08 seconds.

Now to the assumption part. The calculation assumes that the pendulum is in a vacuum, with no air resistance or friction. So, it's like the pendulum is swinging in a perfect world with no pesky forces to mess things up.

Now, if the bob is pulled aside and released, it will start swinging back and forth. At the lowest point of the swing, the bob will have maximum kinetic energy and velocity. So,

(i) The kinetic energy can be calculated using the formula KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass of the bob, and v is the velocity. Plugging in the values, KE = (1/2)(2.2)(v^2). Since we don't know the velocity yet, we can't calculate the exact value of kinetic energy.

(ii) However, we can calculate the velocity using the formula v = √(2gh), where v is the velocity, g is the acceleration due to gravity, and h is the height from which the bob is released. In this case, h is the horizontal distance of 20 cm, which is 0.2 m. Plugging in the values, v = √(2 * 9.8 * 0.2), which comes out to be around 1.98 m/s.

So, at the lowest point of the swing, the velocity of the bob will be approximately 1.98 m/s. As for the exact value of kinetic energy, we'll have to wait for other calculations to determine that. Keep swinging!

To calculate the period of oscillation of a simple 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.

Given:
Length of pendulum (L) = 1.8 m
Mass of bob (m) = 2.2 kg
Acceleration due to gravity (g) = 9.8 m/s^2

Substituting these values into the formula, we get:

T = 2π √(1.8/9.8)

T ≈ 2π √(0.1837)

T ≈ 2π * 0.4289

T ≈ 2.69 s

The assumption made in this calculation is that the pendulum undergoes small angle oscillations, i.e., the angle of displacement of the bob from its equilibrium position is small.

Now, if the bob is pulled aside a horizontal distance of 20 cm (which is equivalent to 0.20 m), and then released, the bob will oscillate back and forth. At the lowest point of the swing, its potential energy will be converted entirely into kinetic energy.

(i) The kinetic energy of the bob at the lowest point of the swing can be calculated using the formula:

KE = 0.5 * m * v^2

Where KE is the kinetic energy, m is the mass of the bob, and v is the velocity.

(ii) The velocity of the bob at the lowest point of the swing can be calculated using the equation:

v = √(2 * g * h)

Where v is the velocity, g is the acceleration due to gravity, and h is the vertical height.

Since the horizontal distance (20 cm) does not affect the vertical height, we can use the same value of h in both equations.

Given:
Horizontal distance (d) = 0.20 m

(i) To find the kinetic energy, we first need to calculate the velocity.

v = √(2 * g * h)

h = length of pendulum (L) - horizontal distance (d)

h = 1.8 - 0.20

h = 1.60 m

v = √(2 * 9.8 * 1.60)

v ≈ √(31.36)

v ≈ 5.60 m/s

KE = 0.5 * m * v^2

KE = 0.5 * 2.2 * (5.60)^2

KE ≈ 0.5 * 2.2 * 31.36

KE ≈ 34.52 J

(ii) The velocity of the bob at the lowest point of the swing is approximately 5.60 m/s.

To calculate the period of oscillation of a simple pendulum, we can use the formula:

T = 2π√(L/g)

where T represents the period of oscillation, L is the length of the pendulum, and g is the acceleration due to gravity. Given that g is 9.8 m/s^2 and the length of the pendulum is 1.8 m, we can substitute these values into the formula:

T = 2π√(1.8/9.8)

Simplifying further:

T = 2π√(0.1837)

T ≈ 2π * 0.4289

T ≈ 2.6948 seconds

So, the period of oscillation of this simple pendulum is approximately 2.6948 seconds.

In this calculation, an assumption that is made is that there is no air resistance acting on the pendulum, and that the pendulum's mass is concentrated at a single point (the bob) at the end of the string. This assumption neglects any friction or drag forces that may be present in the real-world system.

Now, let's move on to the second part of the question.

(i) To calculate the kinetic energy of the bob at the lowest point of the swing, we need to know its velocity at that point. At the lowest point, the bob's potential energy is zero, and all its energy is in the form of kinetic energy.

The velocity of the bob at the lowest point of the swing can be determined using the principle of conservation of energy, which states that the total energy of a closed system remains constant.

The total mechanical energy in the pendulum system is the sum of the gravitational potential energy and the kinetic energy:

E = PE + KE

At the lowest point, the potential energy (PE) is zero, so we have:

E = KE

The total mechanical energy (E) is given by:

E = mgh

where m is the mass of the bob, g is the acceleration due to gravity, and h is the height of the bob above the lowest point. Since the bob is released from a horizontal distance of 20 cm, the height (h) at the lowest point is equal to the length of the pendulum (1.8 m) minus the horizontal distance (0.2 m):

h = 1.8 - 0.2 = 1.6 m

Substituting the values into the equation, we get:

E = (2.2 kg) * (9.8 m/s^2) * (1.6 m) = 34.496 J

Since the total mechanical energy is equal to the kinetic energy (KE) at the lowest point, the kinetic energy is also 34.496 J.

(ii) To find the velocity (v) of the bob at the lowest point, we can use the formula for kinetic energy:

KE = (1/2)mv^2

Rearranging the formula, we get:

v^2 = (2 * KE) / m

Substituting the values:

v^2 = (2 * 34.496 J) / 2.2 kg
v^2 = 31.3596 m^2/s^2

Taking the square root of both sides:

v = √(31.3596) m/s
v ≈ 5.6 m/s

Therefore, the values of (i) the kinetic energy and (ii) the velocity of the bob at the lowest point of the swing are approximately 34.496 J and 5.6 m/s, respectively.