An air puck of mass m1 = 0.24 kg is tied to a string and allowed to revolve in a circle of radius R = 1.4 m on a frictionless horizontal table. The other end of the string passes through a hole in the center of the table, and a mass of m2 = 1.3 kg is tied to it (see the figure below). The suspended mass remains in equilibrium while the puck on the tabletop revolves.

(a) What is the tension in the string?

N

(b) What is the horizontal force acting on the puck?

N

(c) What is the speed of the puck?

m/s

Tension holds the weight

T = 1.3 * 9.81 = 12.75 N

The force on the puck is the tension, 12.75 N

12.75 = m v^2/R

v^2 = 1.4 * 12.75 /.24

v = 8.63 m/s

Why did the air puck cross the table? To get to the other side!

(a) The tension in the string can be found using centripetal force. Since the puck is in equilibrium, the tension in the string is equal to the gravitational force acting on the suspended mass. So the tension is equal to the weight of the suspended mass:

Tension = m2 * g

(b) The horizontal force acting on the puck is provided by the tension in the string. Since there is no friction, the tension provides the only force accelerating the puck towards the center of the circle:

Force = Tension

(c) The speed of the puck can be found using the centripetal force equation:

Force = mass * acceleration

In this case, the centripetal force is provided by the tension in the string:

Tension = mass * (velocity^2 / radius)

Rearranging the equation gives:

velocity^2 = (Tension * radius) / mass

Taking the square root of both sides:

velocity = √((Tension * radius) / mass)

To answer these questions, we need to analyze the forces acting on the system.

(a) Tension in the string:
The tension in the string is equal to the centripetal force acting on the puck. Since the puck is in circular motion, the centripetal force is given by the equation:

F_c = m1 * v^2 / R

where F_c is the centripetal force, m1 is the mass of the puck, v is the speed of the puck, and R is the radius of the circle.

In equilibrium, the tension in the string is also equal to the weight of m2:

T = m2 * g

where T is the tension in the string, m2 is the mass of m2, and g is the acceleration due to gravity.

Setting the centripetal force equal to the tension in the string, we have:

m1 * v^2 / R = m2 * g

Substituting the given values:

0.24 * v^2 / 1.4 = 1.3 * 9.8

Simplifying and solving for v:

v^2 = (1.3 * 9.8 * 1.4) / 0.24

v^2 = 69.0667

v = √69.0667

v ≈ 8.31 m/s

Therefore, the tension in the string is equal to the weight of m2 and is approximately 12.74 N.

(b) Horizontal force on the puck:
The only horizontal force acting on the puck is the tension in the string. Therefore, the horizontal force on the puck is also equal to the tension in the string. From part (a), we found that the tension in the string is approximately 12.74 N.

Therefore, the horizontal force acting on the puck is approximately 12.74 N.

(c) Speed of the puck:
From part (a), we found that the speed of the puck is approximately 8.31 m/s. Therefore, the speed of the puck is approximately 8.31 m/s.

To find the tension in the string in part (a), we can start by analyzing the forces acting on the system.

1. Tension in the string:
Since the suspended mass is in equilibrium, the tension in the string must be equal to the weight of the suspended mass, m2 * g, where g is the acceleration due to gravity. Therefore, we can write the equation:

Tension = m2 * g

2. Horizontal force acting on the puck:
The only horizontal force acting on the puck is the tension in the string. The centripetal force required to keep the puck moving in a circle is provided by this tension. Therefore, the horizontal force acting on the puck is equal to the tension in the string.

Horizontal force = Tension

3. Speed of the puck:
The speed of the puck can be found using the concept of centripetal force. The centripetal force acting on an object moving in a circle of radius R at a speed v is given by the equation:

Centripetal force = mass * velocity^2 / radius

In this case, the centripetal force is provided by the tension in the string. Therefore, we can set up the equation:

Tension = m1 * v^2 / R

Now, let's substitute the given values into these equations to find the answers:

(a) Tension in the string:
Tension = m2 * g = 1.3 kg * 9.8 m/s^2 = 12.74 N

(b) Horizontal force acting on the puck:
Horizontal force = Tension = 12.74 N

(c) Speed of the puck:
Tension = m1 * v^2 / R
12.74 N = 0.24 kg * v^2 / 1.4 m
v^2 = (12.74 N * 1.4 m) / 0.24 kg
v^2 = 74.83 m^2/s^2
v = √(74.83 m^2/s^2) = 8.65 m/s

Therefore:
(a) The tension in the string is 12.74 N.
(b) The horizontal force acting on the puck is 12.74 N.
(c) The speed of the puck is 8.65 m/s.