A "swing" ride at a carnival consists of chairs that are swung in a circle by 17.9 m cables attached to a vertical rotating pole, as the drawing shows. Suppose the total mass of a chair and its occupant is 193 kg. (a) Determine the tension in the cable attached to the chair. (b) Find the speed of the chair.
need the drawing for a solution
or, at least, more data
You are a pirate working for Dread Pirate
Roberts. You are in charge of a cannon
that exerts a force 10000 N on a cannon ball
while the ball is in the barrel of the cannon.
The length of the cannon barrel is 1.7 m and the cannon is aimed at a 38◦angle from the ground.The acceleration of gravity is 9.8 m/s^2.
If Dread Pirate Roberts tells you he wants the ball to leave the cannon with speed V-not = 76 m/s, what mass cannon ball must you use?
Answer in units of kg.
To solve this problem, we can use the principles of circular motion and Newton's laws.
(a) To determine the tension in the cable attached to the chair, we need to consider the forces acting on the system. The two main forces acting on the chair are the weight of the chair and its occupant, and the tension in the cable.
The net force acting towards the center of the circular path is responsible for keeping the chair moving in a circle. This force is given by the equation:
F_net = (mass × velocity^2) / radius
Since the chair is moving in a horizontal circle, the net force is provided by the tension in the cable. Therefore, we can equate the net force to the tension:
F_net = tension
Rearranging the equation, we get:
tension = (mass × velocity^2) / radius
Substituting the known values:
mass = 193 kg and radius = 17.9 m
tension = (193 kg × velocity^2) / 17.9 m
(b) To find the speed of the chair, we can use the equation for centripetal acceleration:
centripetal acceleration = velocity^2 / radius
Rearranging the equation, we get:
velocity^2 = centripetal acceleration × radius
Using the known values:
radius = 17.9 m
velocity^2 = centripetal acceleration × 17.9 m
Now we have two equations:
tension = (193 kg × velocity^2) / 17.9 m
velocity^2 = centripetal acceleration × 17.9 m
To solve the problem, we need to determine the centripetal acceleration. The centripetal acceleration is provided by the gravitational force:
centripetal acceleration = gravitational force / mass
Gravitational force can be calculated using:
gravitational force = mass × gravitational acceleration
Substituting the known values:
mass = 193 kg
gravitational acceleration = 9.8 m/s^2
gravitational force = 193 kg × 9.8 m/s^2
Now we can find the centripetal acceleration:
centripetal acceleration = (193 kg × 9.8 m/s^2) / 193 kg
Substituting this value into the equation for velocity:
velocity^2 = [(193 kg × 9.8 m/s^2) / 193 kg] × 17.9 m
Solving for velocity will give us the speed of the chair.
To determine the tension in the cable attached to the chair, we need to consider the forces acting on the chair. In this case, there are two main forces: the tension in the cable and the force of gravity.
a) First, let's calculate the force of gravity acting on the chair. The force of gravity can be calculated using the equation:
force of gravity = mass * acceleration due to gravity
where the mass is the total mass of the chair and occupant (193 kg) and the acceleration due to gravity is approximately 9.8 m/s^2.
force of gravity = 193 kg * 9.8 m/s^2
force of gravity = 1891.4 N
Now, since the chair is swinging in a circular motion, there is another force acting on it called the centripetal force. The tension in the cable provides this force.
Using the equation for centripetal force:
centripetal force = mass * (velocity)^2 / radius
We have the mass (193 kg) and the radius (17.9 m), but we need to find the velocity to calculate the centripetal force.
b) To find the velocity of the chair, we can use the equation:
velocity = 2πr / time period
Here, the time period is not given, so we need to use the information given in the problem statement.
To find the time period, we can use the equation for the circumference of a circle:
circumference = 2πr
The problem statement states that the chair is swung in a circle with a radius of 17.9 m. Therefore, the circumference is:
circumference = 2π * 17.9 m
circumference = 112.34 m
Since the chair completes one revolution in one time period, the time period is equal to the time it takes to complete one revolution.
Now we can find the velocity:
velocity = 112.34 m / time period
We can now substitute the calculated value of the velocity into the equation for the centripetal force:
centripetal force = 193 kg * (velocity)^2 / 17.9 m
By setting the force of gravity equal to the centripetal force, we can solve for the tension in the cable:
1891.4 N = 193 kg * (velocity)^2 / 17.9 m
Simplifying and rearranging the equation, we get:
(velocity)^2 = (1891.4 N * 17.9 m) / 193 kg
Now that we have the value of (velocity)^2, we can substitute it back into the equation for the centripetal force to find the tension in the cable.
Tension in the cable = 193 kg * (velocity)^2 / 17.9 m
After calculating the velocity and substituting it into the equation, we can find the tension in the cable.