A small object with a mass of m = 615 g is whirled at the end of a rope in a vertical circle with a radius of r = 123 cm.
When it is at the location shown, (mid-height), its speed is v = 5.60 m/s. Determine the tension in the rope.
Calculate the magnitude of the total force acting on the mass at that location.
M = 615g = 0.615kg
R = 123cm = 0.123m
V = 5.60 ms^-1
Tension = m * v / r
0.615 / 0.123 * 5.60 = 28 Nm^-1
I think R should be 1.23m there shredder.
To determine the tension in the rope, we can use the centripetal force equation: F_c = (mv^2) / r, where F_c is the centripetal force, m is the mass, v is the velocity, and r is the radius.
Step 1: Convert the mass to kilograms:
m = 615 g = 0.615 kg
Step 2: Convert the radius to meters:
r = 123 cm = 1.23 m
Step 3: Calculate the centripetal force:
F_c = (0.615 kg)(5.60 m/s)^2 / 1.23 m
Step 4: Calculate the tension in the rope:
The tension in the rope is equal to the centripetal force, so the tension is:
Tension = F_c = (0.615 kg)(5.60 m/s)^2 / 1.23 m
To calculate the magnitude of the total force acting on the mass at that location, we need to consider both the gravitational force and the tension force.
Step 1: Calculate the gravitational force:
The gravitational force is given by the equation: F_g = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_g = (0.615 kg)(9.8 m/s^2)
Step 2: Calculate the total force:
The total force is the vector sum of the tension force and the gravitational force (since they are in opposite directions at this point):
Total force = Tension - Gravitational force
Substitute the values calculated above into this equation to find the magnitude of the total force acting on the mass at that location.
To calculate the tension in the rope and the magnitude of the total force acting on the mass at the given location in the vertical circle, we can use the concept of centripetal force. The centripetal force is the net force directed toward the center of the circular path, which keeps the object moving in a circle.
Step 1: Find the gravitational force acting on the object.
The gravitational force acting on an object can be calculated using the formula:
F_gravity = m * g
where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s²).
Given: m = 615 g = 0.615 kg and g = 9.8 m/s²
Substituting the values into the formula:
F_gravity = 0.615 kg * 9.8 m/s²
F_gravity ≈ 6.027 N
Step 2: Determine the tension in the rope.
At the mid-height location, the tension in the rope provides the centripetal force responsible for keeping the object in a circular motion.
The centripetal force in this case is given by the formula:
F_centripetal = (m * v²) / r
where m is the mass of the object, v is its velocity, and r is the radius of the circular path.
Given: m = 0.615 kg, v = 5.60 m/s, and r = 123 cm = 1.23 m
Substituting the values into the formula:
F_centripetal = (0.615 kg * (5.60 m/s)²) / 1.23 m
F_centripetal ≈ 16.859 N
Therefore, the tension in the rope is approximately 16.859 N.
Step 3: Calculate the magnitude of the total force.
The total force acting on the object at the mid-height location is the vector sum of the gravitational force and the tension in the rope.
The magnitude of the total force can be found using vector addition:
F_total = √(F_gravity² + F_centripetal²)
Substituting the calculated values:
F_total = √((6.027 N)² + (16.859 N)²)
F_total ≈ √(36.324 N² + 284.168 N²)
F_total ≈ √320.492 N²
F_total ≈ 17.9 N
Therefore, the magnitude of the total force acting on the mass at that location is approximately 17.9 N.