An air puck of mass 0.289 kg is tied to a string

and allowed to revolve in a circle of radius
0.51 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 0.98 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

hangingmass*g=tension in string.

goodness.

To find the tension in the string, we can use the concept of centripetal force.

The tension in the string is equal to the centripetal force required to keep the air puck in circular motion.

The centripetal force is given by the formula:

Fc = (m * v^2) / r

Where:
Fc = centripetal force
m = mass of the air puck
v = velocity of the air puck
r = radius of the circle

Since the air puck is in equilibrium, the centripetal force is balanced by the tension in the string and the force due to gravity:

Tension + Force due to gravity = Fc

The force due to gravity is given by:

Force due to gravity = m * g

Where:
g = acceleration due to gravity

Substituting the values into the equation:
Tension + (m * g) = (m * v^2) / r

Rearranging the equation to solve for the tension:
Tension = (m * v^2) / r - (m * g)

Plugging in the values given:
m = 0.289 kg
r = 0.51 m
g = 9.8 m/s^2

Now we need to find the velocity of the air puck. Since the mass tied to the string is in equilibrium, its weight is balanced by the tension in the string:

Tension = Force due to gravity

Solving for the tension:
Tension = m * g

Plugging in the values given:
m = 0.98 kg
g = 9.8 m/s^2

Now we can plug the values into the equation for tension and find the answer:

Tension = (0.289 kg * v^2) / 0.51 m - (0.98 kg * 9.8 m/s^2)

Now let's calculate it:

To find the tension in the string, we need to consider the forces acting on the system.

1. Tension in the string:
The tension in the string can be determined by considering the forces acting on the puck. Since the puck is moving in a circle, there must be a centripetal force providing the necessary inward acceleration. The tension in the string provides this force. Therefore, the tension in the string is equal to the centripetal force.

2. Centripetal force:
The centripetal force required to keep an object moving in a circular path is given by the formula:

F_c = m * a_c

where F_c is the centripetal force, m is the mass of the object, and a_c is the centripetal acceleration.

3. Centripetal acceleration:
The centripetal acceleration of an object moving in a circle of radius r and with a velocity v is given by the formula:

a_c = v^2 / r

where a_c is the centripetal acceleration, v is the velocity of the object, and r is the radius of the circle.

4. Velocity of the puck:
The velocity of the puck can be found using the concept of angular velocity. Since the puck is tied to the string, it will have a constant angular velocity. The angular velocity (ω) is related to the linear velocity (v) by the equation:

v = ω * r

where v is the linear velocity, ω is the angular velocity, and r is the radius of the circle.

5. Angular velocity:
The angular velocity of the puck can be found by considering the equilibrium of the system. The hanging mass is in equilibrium, meaning the tension in the string is equal to the weight of the hanging mass. Therefore, we can use the weight of the hanging mass to find the angular velocity.

Now, let's calculate the tension in the string:

Step 1: Calculate the angular velocity (ω):
The weight of the hanging mass is given by:

w = m * g

where w is the weight, m is the mass of the hanging mass, and g is the acceleration due to gravity.

Substituting the given values, we have:

w = (0.98 kg) * (9.8 m/s^2) = 9.604 N

Since the weight of the hanging mass is equal to the tension in the string, we have:

Tension = Weight = 9.604 N.

Therefore, the tension in the string is 9.604 N.