4.A 30.8-kg child swings on a rope with a length of 6.29 m that is hanging from a tree. At the bottom of the swing, the child is moving at a speed of 4.2 m/s. What is the tension in the rope?
M*(g + V^2/L)
To find the tension in the rope, we can use the concept of centripetal force. The tension in the rope provides the centripetal force required to keep the child moving in a circular path.
The centripetal force can be calculated using the formula:
F = (m * v^2) / r
Where:
- F is the centripetal force
- m is the mass of the child
- v is the velocity of the child
- r is the radius of the circular path (which is equal to the length of the rope)
Plugging in the given values:
m = 30.8 kg
v = 4.2 m/s
r = 6.29 m
F = (30.8 kg * (4.2 m/s)^2) / 6.29 m
Now, let's calculate the value:
F ≈ 20.50 N
Therefore, the tension in the rope is approximately 20.50 N.
To find the tension in the rope, we can use the principle of conservation of mechanical energy. The mechanical energy is conserved in this system because there are no external forces acting on the child-rope system (assuming negligible air resistance). The energy is transformed between kinetic energy (when the child is moving) and gravitational potential energy (when the child is at the highest point of the swing).
The formula for gravitational potential energy is given by:
PE = m * g * h
Where:
PE is the gravitational potential energy
m is the mass of the child (30.8 kg)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the height of the swing from the lowest point to the highest point (6.29 m)
The kinetic energy is given by:
KE = (1/2) * m * v^2
Where:
KE is the kinetic energy
m is the mass of the child (30.8 kg)
v is the velocity of the child at the bottom of the swing (4.2 m/s)
At the highest point of the swing, all the energy is in the form of gravitational potential energy, so we have:
PE = KE
Now we can substitute the formulas and solve for the tension in the rope.
PE = KE
m * g * h = (1/2) * m * v^2
Canceling out the mass (m) on both sides, we have:
g * h = (1/2) * v^2
Substituting the given values:
9.8 m/s^2 * 6.29 m = (1/2) * (4.2 m/s)^2
Simplifying:
61.042 m^2/s^2 = 44.1 m^2/s^2
Now, we can find the tension in the rope using the formula:
Tension = m * g + m * v^2 / r
Where:
Tension is the tension in the rope
m is the mass of the child (30.8 kg)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
v is the velocity of the child at the bottom of the swing (4.2 m/s)
r is the length of the rope (6.29 m)
Substituting the values:
Tension = (30.8 kg * 9.8 m/s^2) + (30.8 kg * (4.2 m/s)^2) / 6.29 m
Calculating:
Tension = 301.84 N + 241.02 N
So, the tension in the rope is approximately 542.86 N.