you are walking into a circus tent and you notice the circus ringleader twirling a 150 g gold pocket watch from a chain that is 40.0 cm long. the period of rotation is about one second.
a) determine the tension in the gold chain when the watch is at the top
b) determine the tension in the gold chain when the watch is at the bottom
To determine the tension in the gold chain when the watch is at the top, we can use the centripetal force equation:
Tension = (mass × velocity²) / radius
a) First, let's find the velocity of the gold pocket watch at the top. We know that the period of rotation is one second, which means it completes one full revolution in that time. Since the radius of the circular motion is given by the length of the chain, which is 40.0 cm, we can calculate the circumference using the formula:
Circumference = 2π × radius
Substituting the value of the radius, we get:
Circumference = 2π × 40.0 cm
Next, to find the velocity, we divide the circumference by the period:
Velocity = Circumference / Period
Substituting the values, we get:
Velocity = (2π × 40.0 cm) / 1 s
Now that we have the velocity, let's calculate the tension in the chain.
Tension = (mass × velocity²) / radius
Given that the mass of the pocket watch is 150 g (or 0.150 kg), we can substitute the values:
Tension = (0.150 kg × (2π × 40.0 cm / 1 s)²) / 40.0 cm
By simplifying the equation, we get the tension.
b) To determine the tension in the gold chain when the watch is at the bottom, we repeat the same steps as above, but with a slight modification.
The only difference is that at the bottom of the circular motion, the tension in the chain needs to counterbalance not only the centripetal force but also the weight of the pocket watch.
So, we need to consider the additional weight due to gravity. The weight is given by:
Weight = mass × acceleration due to gravity
Substituting the values, we get:
Weight = 0.150 kg × 9.8 m/s²
Now, we can calculate the total tension required at the bottom using the equation:
Tension = (mass × velocity²) / radius + weight
Substituting the values, we get:
Tension = (0.150 kg × (2π × 40.0 cm / 1 s)²) / 40.0 cm + (0.150 kg × 9.8 m/s²)
By simplifying the equation, we can find the tension in the chain when the watch is at the bottom.