State the law of conservation of energy - Leaving Cert Physics - Question a - 2013

To calculate the height (h), use the formula:

$h = L(1 - ext{cos} heta)$

where:

• L = 8 m (length of the pendulum)
• \theta = 30°

Substituting the values, we find:

$h = 8 - 8 imes ext{cos} 30 = 1.07 ext{ m}$

Next, we calculate the potential energy (E) gained using:

$E = mgh$

where:

• m = 6 kg (mass of the bob)
• g = 9.8 m/s² (acceleration due to gravity)

Substituting the values:

$E = 6 imes 9.8 imes 1.07 = 63 ext{ J}$

The maximum kinetic energy (KE) is equal to the potential energy at the top of the pendulum swing:

$ext{KE} = E = 63 ext{ J}$

Using the kinetic energy formula:

ext{KE} = rac{1}{2} mv^2

Rearranging for v gives:

v = ext{sqrt}igg(rac{2 imes ext{KE}}{m}igg)

Substituting the values:

v = ext{sqrt}igg(rac{2 imes 63}{6}igg) = 4.58 ext{ m/s}

To find the force (F) applied, use the work-energy principle:

$W = Fd$

where W is the work done. The work done is equal to the change in kinetic energy:

$W = KE_{initial} - KE_{final}$

At position A, the initial kinetic energy is 63 J, and after stopping, it’s 0 J. The distance (d) to stop is 0.005 m:

F = rac{W}{d} = rac{63 - 0}{0.005} = 12604.3 ext{ N}