A gymnast of mass 63.0 {\rm kg} hangs from a vertical rope attached to the ceiling. You can ignore the weight of the rope and assume that the rope does not stretch.Calculate the tension T in the rope if the gymnast climbs up the rope with an upward acceleration of magnitude 0.500 {\rm m/s^2}.

T=mg+ma

mg = weight = 63.o*9.81
ma= 63*.5
then add to get T

T= 496.368 N

= 496 N

To calculate the tension in the rope, we need to consider the forces acting on the gymnast.

1. The force of gravity (weight) acting downward:
Fg = m * g
= 63.0 kg * 9.8 m/s^2
= 617.4 N

2. The force of tension in the rope acting upward (T):
We can use Newton's second law of motion to find the tension:
Fnet = m * a
Where Fnet is the net force, m is the mass, and a is the acceleration.

Since the gymnast is climbing up the rope with an upward acceleration, the net force is the sum of the tension force (T) and the force of gravity (Fg):
Fnet = T - Fg

Therefore,
T - Fg = m * a
T = m * a + Fg

Substituting the given values:
T = 63.0 kg * 0.500 m/s^2 + 617.4 N
T = 31.5 N + 617.4 N
T ≈ 648.9 N

Therefore, the tension in the rope is approximately 648.9 N.

To calculate the tension in the rope, we need to consider the forces acting on the gymnast. In this case, there are only two forces involved: the gravitational force acting downward and the tension force exerted by the rope acting upward.

1. Start by calculating the gravitational force acting on the gymnast using the equation: F_gravity = m * g
- Here, m represents the mass of the gymnast, which is given as 63.0 kg.
- g represents the acceleration due to gravity, which is approximately 9.8 m/s^2.

F_gravity = (63.0 kg) * (9.8 m/s^2) = 617.4 N

2. Determine the net force acting on the gymnast using Newton's second law of motion: F_net = m * a
- Here, m represents the mass of the gymnast, which is 63.0 kg.
- a represents the acceleration of the gymnast while climbing up the rope.
- In this case, the acceleration is given as 0.500 m/s^2.

F_net = (63.0 kg) * (0.500 m/s^2) = 31.5 N

3. The net force acting on the gymnast is equal to the difference between the tension force and the gravitational force.
F_net = T - F_gravity

4. Rearrange the equation to solve for T:
T = F_net + F_gravity

T = (31.5 N) + (617.4 N) = 648.9 N

Therefore, the tension in the rope is approximately 648.9 N.