Question

A pendulum swings through an arc of 90.0° (45.0° on either side of the vertical). The mass of the bob is 4.10 kg and the length of the suspending cord is 1.70 m.

(a) Find the tension in the cord at the end points of the swing.
  

(b) Find the velocity of the bob as it passes its lowest point and the tension in the cord at this point.

velocity	=
tension	=

Answer 1

A pendulum swings through an arc of 90.0° (45.0° on either side of the vertical). The mass of the bob is 4.10 kg and the length of the suspending cord is 1.70 m.



As the pendulum swings downward from either end point, its velocity increases. The maximum velocity occurs at the lowest point of the motion. At the end points, the velocity = 0. At the lowest point, the kinetic energy is maximum. At the highest point, the potential energy is maximum.

Change of KE = change of PE
Change of KE = ½ * mass * (vf^2 – vi^2), vi = 0 m/s
Change of KE = ½ * 4.10 * vf^2

Change of PE = mass * g * ∆ height
∆ height = length of cord – length of cord * cos θ
∆ height = (1.70 –1.70 * cos 45°) = .497

½ * 4.10 * vf^2 = 4.10 * 9.81 * .497
½ * vf^2 = 19.26/4.10
vf^2 =4.69*2
vf = 3.06m/s

The bob is moving in a circular arc. The centripetal force is directed toward the center of the circle. The centripetal force has 2 components. The tension in the cord is pulling the bob toward the center and the component of weight parallel to the cord is pulling the bob away from the center. This could be called the radial component of the weight.


Radial component of weight = mass * g * cos θ
Sum of the forces = Centripetal force
T – mass * g * cos θ = mass * v^2 ÷ radius

mass = 4.10 kg
radius = length of cord = 1.70 m

At the end points, θ = 45°, v = 0

T – 4.10 * 9.8 * cos 45° = 4.10 * 0^2 ÷ 1.70

T – 4.10 * 9.8 * cos 45° = 0
T = 4.10 * 9.8 * cos 45° = 28.41 N

At lowest point, θ = 0°, v = 3.06m/s

T – 4.10 * 9.8 * cos 0° = 4.10 * 3.06^2 ÷ 1.70
T = (4.10 * 9.8 * cos 0°) - (4.10 * 3.06^2 ÷ 1.70)
T =17.69 N

a) Find the tension in the cord at the end points of the swing.
T = 28.41 N


b) Find the velocity of the bob as it passes its lowest point and the tension in the cord at this point.
velocity = 3.06m/s
tension = 17.69 N