A rubber stopper with a mass of 50.0 g attached to a string is being rotated in a vertical circle with a radius of 0.72 m at a constant speed of 7.3 m/s. (/2)
What is the tension in the string when the stopper is at the top of the circle?
What is the tension in the string when the stopper is at the bottom of the circle.
weight down = m g
tension due to centipedal acceleration = m v^2/R
at top T = m (v^2/R - g)
at bottom T = m (v^2/R + g)
To find the tension in the string at the top and bottom of the circle, we need to consider the forces acting on the rubber stopper.
At the top of the circle:
When the stopper is at the top of the circle, the net force acting on it is directed towards the center of the circle. The gravitational force and the tension force add up to provide this net force.
1. Find the gravitational force:
The gravitational force acting on the stopper is given by the equation:
F_gravity = m * g
where m is the mass of the stopper (50.0 g = 0.050 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²).
F_gravity = 0.050 kg * 9.8 m/s² = 0.49 N
2. Find the net force:
The net force acting on the stopper at the top of the circle is given by:
F_net = m * a
where m is the mass of the stopper and a is the centripetal acceleration.
The centripetal acceleration is given by:
a = v² / r
where v is the speed of the stopper (7.3 m/s) and r is the radius of the circle (0.72 m).
a = (7.3 m/s)² / 0.72 m = 74.60 m/s²
F_net = 0.050 kg * 74.60 m/s² = 3.73 N
3. Find the tension in the string at the top:
To find the tension in the string, we subtract the gravitational force from the net force, as the tension force is in the opposite direction to the gravitational force.
Tension at the top = F_net - F_gravity
Tension at the top = 3.73 N - 0.49 N = 3.24 N
So, the tension in the string when the stopper is at the top of the circle is 3.24 N.
At the bottom of the circle:
When the stopper is at the bottom of the circle, the net force acting on it is directed towards the center of the circle. The gravitational force and the tension force add up to provide this net force.
1. Find the gravitational force (same as above): F_gravity = 0.49 N
2. Find the net force (same as above): F_net = 3.73 N
3. Find the tension in the string at the bottom:
To find the tension in the string, we add the gravitational force to the net force, as the tension force is in the same direction as the gravitational force.
Tension at the bottom = F_net + F_gravity
Tension at the bottom = 3.73 N + 0.49 N = 4.22 N
So, the tension in the string when the stopper is at the bottom of the circle is 4.22 N.
To find the tension in the string at the top and bottom of the circle, we need to consider the forces acting on the rubber stopper. At the top of the circle, the tension in the string will be responsible for providing the required centripetal force to keep the stopper moving in a circle. At the bottom of the circle, the tension will need to provide both the centripetal force and counteract the force due to the weight of the stopper.
To find the tension at the top of the circle, we'll begin by calculating the centripetal force. The centripetal force (Fc) is given by the equation:
Fc = (m * v^2) / r
Where:
m = mass of the stopper
v = velocity of the stopper
r = radius of the circle
Plugging in the values:
m = 50.0 g = 0.050 kg (converting grams to kilograms)
v = 7.3 m/s
r = 0.72 m
Substituting the values into the equation:
Fc = (0.050 kg * (7.3 m/s)^2) / 0.72 m
Fc ≈ 0.497 N
Since the tension in the string at the top of the circle provides the centripetal force, the tension is equal to the centripetal force:
Tension at the top = 0.497 N
To find the tension at the bottom of the circle, we need to consider the force due to gravity. The force due to gravity (mg) is given by the equation:
mg = (m * g)
Where:
m = mass of the stopper
g = acceleration due to gravity (approximately 9.8 m/s^2)
Plugging in the values:
m = 50.0 g = 0.050 kg (converting grams to kilograms)
g = 9.8 m/s^2
Substituting the values into the equation:
mg = (0.050 kg * 9.8 m/s^2)
mg ≈ 0.490 N
Since the tension in the string at the bottom of the circle needs to provide both the centripetal force and counteract the force due to gravity, we can calculate the tension by summing the forces:
Tension at the bottom = Fc + mg = 0.497 N + 0.490 N
Tension at the bottom ≈ 0.987 N
Therefore, the tension in the string when the stopper is at the top of the circle is approximately 0.497 N, and the tension in the string when the stopper is at the bottom of the circle is approximately 0.987 N.