An air puck of mass 0.23 kg is tied to a string and allowed to revolve in a circle of radius 1.0 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 0.9 kg is tied to it. 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 force causing the centripetal acceleration on the puck?
N
(c) What is the speed of the puck?
m/s
. An air puck of mass 0.250 kg is tied to a string
and allowed to revolve in a circle of radius
1.00m 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 1.00 kg is
tied to it (Fig. 4). The suspended mass remains
in equilibrium while the puck on the tabletop
revolves. What are (a) the tension in the string,
(b) the force exerted by the string on the puck,
and (c) the speed of the puck?
To answer these questions, we need to apply the principles of circular motion and equilibrium.
(a) Tension in the string:
In this scenario, the tension in the string is the force that keeps both the puck and the suspended mass in equilibrium. This tension can be calculated by considering the force balance on the suspended mass.
Since the suspended mass is in equilibrium, the tension in the string must be equal to the weight of the suspended mass. The weight of the suspended mass can be calculated using the formula:
Weight = mass * gravity
where:
mass = 0.9 kg (given)
gravity = 9.8 m/s^2 (acceleration due to gravity)
Weight = 0.9 kg * 9.8 m/s^2 = 8.82 N
Therefore, the tension in the string is 8.82 N.
(b) Force causing the centripetal acceleration on the puck:
The centripetal force required to keep the puck moving in a circular path is provided by the tension in the string. It can be calculated using the formula:
Centripetal force = mass * (centripetal acceleration)
The centripetal acceleration can be calculated using the formula:
Centripetal acceleration = (velocity^2) / radius
However, we don't have the velocity of the puck yet, so we'll need to calculate it before proceeding.
(c) Speed of the puck:
To calculate the speed of the puck, we can use the concept of equilibrium. In equilibrium, the net force acting on an object is zero. In this case, the net force on the puck is the tension in the string, since there is no friction on the tabletop.
The force balance equation for the puck can be written as:
Tension = (mass of the puck) * (centripetal acceleration)
Rearranging the equation, we have:
Centripetal acceleration = Tension / (mass of the puck)
Substituting the known values:
Centripetal acceleration = 8.82 N / 0.23 kg = 38.3 m/s^2
Now, we can use this centripetal acceleration to find the velocity (speed) of the puck using the equation:
Centripetal acceleration = (velocity^2) / radius
Rearranging the equation, we have:
Velocity^2 = (Centripetal acceleration) * radius
Velocity^2 = 38.3 m/s^2 * 1.0 m = 38.3 m^2/s^2
Taking the square root of both sides:
Velocity = √(38.3 m^2/s^2) = 6.19 m/s
Therefore, the speed of the puck is approximately 6.19 m/s.
To summarize the answers:
(a) The tension in the string is 8.82 N.
(b) The force causing the centripetal acceleration on the puck is the tension in the string, which is 8.82 N.
(c) The speed of the puck is approximately 6.19 m/s.