an air puck of mass 0.347 kg is tied to a string and allowed to revolve in a circle of radius 1.09 m on a horizontal, frictionless table. The other end of the string passes through a hole in the center of the table and a mass of 1.48 kg is tied to it. The suspended mass remains in equilibrium while the puck revolves. What is the tension in the string?
Since the string tension supports the 1.48 kg mass that dangles from it, in equilibrium,
T = M g = 14.50 N
You don't need the information on the air puck on the table to compute the tension, but you could use the tension to compute the required speed of the air puck.
Kind of a trick question. You have to know what information to use and what to ignore. Just like life.
To find the tension in the string, we can consider the forces acting on the puck.
1. Centripetal Force: The puck is moving in a circle, so it experiences a centripetal force directed towards the center of the circle. This force is given by the equation F_c = (m * v^2) / r, where m is the mass of the puck, v is its velocity, and r is the radius of the circle.
2. Tension Force: The tension in the string is the force exerted by the string on the puck. Since the puck is in equilibrium, the tension force must balance the centripetal force.
Let's calculate the tension in the string step by step:
Step 1: Calculate the centripetal force.
We know the mass of the puck (m = 0.347 kg) and the radius of the circular path (r = 1.09 m). However, we don't know the velocity. To find it, we can use the concept of centripetal acceleration.
The centripetal acceleration is given by the equation a_c = v^2 / r. Rearranging the equation, we get v^2 = a_c * r.
Since the puck is in equilibrium, the centripetal acceleration can be calculated using the acceleration due to gravity (g) and the mass of the hanging mass (m_hanging). The acceleration due to gravity is given by g = 9.8 m/s^2.
The centripetal acceleration is equal to g * m_hanging, so a_c = g * m_hanging.
Plugging in the values, we can calculate the centripetal acceleration:
a_c = 9.8 m/s^2 * 1.48 kg
Step 2: Calculate the velocity of the puck.
Now that we know the centripetal acceleration, we can find the velocity using the equation v^2 = a_c * r.
v^2 = a_c * r = (9.8 m/s^2 * 1.48 kg) * 1.09 m
Step 3: Calculate the centripetal force.
Now that we have the velocity, we can calculate the centripetal force using the formula F_c = (m * v^2) / r.
F_c = (0.347 kg * v^2) / r = (0.347 kg * (9.8 m/s^2 * 1.48 kg) * 1.09 m) / 1.09 m
Step 4: Calculate the tension in the string.
Since the suspended mass remains in equilibrium, the tension force in the string balances the centripetal force.
Therefore, the tension in the string is equal to the centripetal force:
Tension = F_c
Now you can plug in the values into the equation to find the tension in the string.