A student spins a 1.2kg rock around above her head horizontally in an experiment which aims to model how satellites orbit the earth. The student counts 3 orbits per second when the string produces an orbit radius of 35cm.
(a) Calculate the centripetal force acting on the rock.
(b) Describe the effects of two real forces acting on the revolving rock in this situation.
Fc = m v^2/r
circumference = C = 2 pi r = 2 pi * .35
speed = v = circumference/time = 2 pi *35/(1/3) meters/second
so
Fc = 1.2 * (6pi*.35)^2/.35 N
other force is gravity down
1.2*9.8 N
tan angle down from horizontal = 1.2*9.8 /Fc
(a) To calculate the centripetal force acting on the rock, we can use the formula for centripetal force:
F = m * v^2 / r
Where:
F is the centripetal force,
m is the mass of the rock (1.2 kg),
v is the linear velocity of the rock, and
r is the radius of the orbit.
To find the linear velocity, we can use the given information that there are 3 orbits per second and the orbit radius is 35 cm.
First, let's convert the orbit radius from centimeters to meters:
r = 35 cm = 0.35 m
The linear velocity can be found by multiplying the circumference of the orbit by the number of orbits per second:
v = 2 * π * r * n
Where n is the number of orbits per second.
Plugging in the values:
v = 2 * 3.14 * 0.35 * 3 = 6.6156 m/s
Now, we can calculate the centripetal force:
F = 1.2 kg * (6.6156 m/s)^2 / 0.35 m
F = 1.2 * 43.7 / 0.35 = 152.9143 N
Therefore, the centripetal force acting on the rock is approximately 152.9143 N.
(b) In this situation, there are two real forces acting on the revolving rock:
1. Centripetal Force: This is the force acting towards the center of the circular orbit, allowing the rock to continue moving in a circle. It is provided by the tension in the string and its value has been calculated in part (a) as approximately 152.9143 N.
2. Gravity: The force of gravity is also acting on the rock, pulling it downward towards the center of the Earth. However, because the rock is orbiting in a horizontal plane, the force of gravity does not affect the shape of the orbit. It does, however, affect the weight of the rock. The weight of the rock remains the same but is counteracted by the centripetal force, resulting in a net force of zero in the vertical direction.
In summary, the two real forces acting on the revolving rock in this situation are the centripetal force (provided by the tension in the string) and the force of gravity (being counteracted by the centripetal force).
To calculate the centripetal force acting on the rock, we can use the formula for centripetal force:
Fc = (m * v^2) / r
Where:
- Fc is the centripetal force
- m is the mass of the rock (1.2 kg)
- v is the velocity of the rock
- r is the radius of the orbit (35 cm or 0.35 m)
To find the velocity of the rock, we need to know the circumference of the orbit. Since the radius is given, we can use the formula for the circumference of a circle:
C = 2 * π * r
Let's calculate the velocity first:
v = C / T
Where:
- v is the velocity
- C is the circumference
- T is the time for one orbit (1/3 seconds)
Substituting the values:
C = 2 * π * 0.35 m = 2.20 m (approximately)
T = 1/3 seconds
v = 2.20 m / (1/3) s = 2.20 * 3 m/s = 6.60 m/s
Now, we can substitute the values of mass (m = 1.2 kg), velocity (v = 6.60 m/s), and radius (r = 0.35 m) into the centripetal force formula:
Fc = (1.2 kg * (6.60 m/s)^2) / 0.35 m
Calculating this equation gives us:
Fc ≈ 22.286 N
Therefore, the centripetal force acting on the rock is approximately 22.286 N.
Moving on to part (b) of the question, let's describe the effects of two real forces acting on the revolving rock.
1. Tension Force from the String: The tension force in the string provides the centripetal force required to keep the rock in circular motion. It pulls the rock towards the center of the orbit, preventing it from moving in a straight line. If the tension force suddenly disappears, the rock will move off in a straight line tangent to its orbit.
2. Gravitational Force: The gravitational force attracts the rock toward the center of the Earth. Although the gravitational force is always present, in this situation, it does not have a significant effect on the rock's orbit since the centripetal force provided by the tension force exactly balances it. However, if the gravitational force were stronger or weaker, it would affect the orbital radius of the rock. A stronger gravitational force would cause the rock to have a smaller radius, while a weaker gravitational force would cause it to have a larger radius.