An air puck of mass 0.278 kg is tied to a string

and allowed to revolve in a circle of radius
1.25 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 0.45 kg
is tied to it. The suspended mass remains in
equilibrium while the puck revolves.
What is the tension in the string? The
acceleration due to gravity is 9.8 m/s
2
.
Answer in units of N

To find the tension in the string, we need to sum the forces acting on the suspended mass in equilibrium.

First, let's consider the forces acting on the suspended mass:
1. The force of gravity, which is equal to the mass of the suspended mass (0.45 kg) multiplied by the acceleration due to gravity (9.8 m/s^2). Therefore, the force of gravity is (0.45 kg)(9.8 m/s^2) = 4.41 N. However, this force is directed downward, so we'll consider it with a negative sign.

Next, let's consider the forces acting on the air puck:
1. The normal force, which is equal to the weight of the air puck (mass of the air puck multiplied by the acceleration due to gravity). Since the table is horizontal and frictionless, the normal force is equal in magnitude and opposite in direction to the force of gravity on the air puck. Therefore, the normal force is (-0.278 kg)(9.8 m/s^2) = -2.72 N. Again, we'll consider this force with a negative sign because it is directed upward.

Since the air puck is revolving in a circle of radius 1.25 m, there must be a centripetal force acting on it. The centripetal force is equal to the mass of the air puck multiplied by its centripetal acceleration.

The centripetal acceleration of an object moving in a circle of radius r with a speed v can be found using the equation a = v^2 / r.

In this case, we don't know the speed of the air puck directly. However, we can find it indirectly using the information about the suspended mass.

Since the suspended mass is in equilibrium, the magnitudes of the centripetal force on the air puck and the force of gravity on the suspended mass must be equal.

Therefore, we can equate the centripetal force to the force of gravity on the suspended mass:

(mass of the air puck)(centripetal acceleration) = (mass of the suspended mass)(acceleration due to gravity)

Solving for the centripetal acceleration, we get:

centripetal acceleration = (mass of the suspended mass)(acceleration due to gravity) / (mass of the air puck)

Plugging in the given values, the centripetal acceleration is (0.45 kg)(9.8 m/s^2) / (0.278 kg) = 15.99 m/s^2.

Now, we can find the centripetal force by multiplying the mass of the air puck by the centripetal acceleration:

centripetal force = (mass of the air puck)(centripetal acceleration) = (0.278 kg)(15.99 m/s^2) = 4.45 N.

To get the tension in the string, we need to sum the forces acting on the suspended mass in equilibrium:

Tension in the string = force of gravity on the suspended mass + centripetal force

Tension in the string = (-4.41 N) + (4.45 N) = 0.04 N.

Therefore, the tension in the string is 0.04 N.