A pendulum bob of mass 0.5kg hanging from a vertical string of length 20m attached to a fixed point is pulled to one side with the string taut until it makes an angle 30° to the vertical. Calculate the work done on the bob?

Question

Isabella Alvarez · Accepted Answer

Using Law of conservation of energy and assuming that pendulum is ideal and no energy is lost in friction,

Work done on the bob#=# Increase in potential Energy of bob

Increase in potential Energy of bob#=mgDeltah#
where #m# is mass of bob, #g=9.81\ ms^-2# is acceleration due to gravity and #Deltah# is change in height of bob.
As shown in the figure, #Deltah=L-Lcostheta#. Inserting given values we get

Work done on the bob#=0.5xx9.81(20-20\ cos30^@)#
#=>#Work done on the bob#=0.5xx9.81xx20(1- sqrt3/2)=13.1\ J#

Parker Abbott · Answer

undefined The work done on the bob can be calculated using the formula:

$$ W = mgh $$

where:
- $ m $ = mass of the bob (0.5 kg)
- $ g $ = acceleration due to gravity (9.8 m/s^2)
- $ h $ = height of the bob above its lowest point (which is equal to the vertical displacement of the bob)

The vertical displacement of the bob can be calculated using trigonometry:

$$ h = L(1 - \cos(\theta)) $$

where:
- $ L $ = length of the string (20 m)
- $ \theta $ = angle made by the string with the vertical (30°)

After calculating $ h $, plug the values into the formula for work:

$$ W = (0.5 \, \text{kg})(9.8 \, \text{m/s}^2)(20 \, \text{m})(1 - \cos(30°)) $$

$$ W \approx 0.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 20 \, \text{m} \times (1 - \cos(30°)) $$

$$ W \approx 0.5 \times 9.8 \times 20 \times (1 - \cos(30°)) $$

$$ W \approx 0.5 \times 9.8 \times 20 \times (1 - \frac{\sqrt{3}}{2}) $$

$$ W \approx 0.5 \times 9.8 \times 20 \times (1 - 0.866) $$

$$ W \approx 0.5 \times 9.8 \times 20 \times 0.134 $$

$$ W \approx 13.176 \, \text{J} $$

So, the work done on the bob is approximately 13.176 Joules.