A pendulum consists of a sphere of mass 1.5 kg attached to a light cord of length 10.9 m as in the figure below. The sphere is released from rest when the cord makes a 58 degree angle with the vertical, and the pivot at P is frictionless. The acceleration of gravity is 9.8 m/s^2 . Find the speed of the sphere when it is at the lowest point. Answer in units of m/s

Question

What is the tension in the cord?

Answer 1

First, we need to find the vertical height from the initial position to the lowest point.

Let's denote the height as h, the length of the cord, as L (10.9 m), and the angle between the cord and the vertical as θ (58 degrees).

We can use the following equation to find h:

h = L - L*cos(θ)
h = 10.9 - 10.9*cos(58)
h = 10.9 - 10.9*0.5299
h = 10.9 - 5.7762
h = 5.1238 m

Next, we can find the potential energy change and consequent kinetic energy gain when the sphere reaches the lowest point.

Potential energy change (PE) is given by:

PE = m * g * h
PE = 1.5*9.8*5.1238
PE = 75.3303 J

At the lowest point, all the potential energy turns into kinetic energy. So, we can use the kinetic energy formula to find the speed (v) of the sphere:

KE = 0.5*m*v^2
75.3303 = 0.5*1.5*v^2

Now, we can solve for v:

v^2 = (75.3303*2)/1.5
v^2 = 100.4404

v = sqrt(100.4404)
v = 10.022 m/s

So, the speed of the sphere at the lowest point is 10.022 m/s.

Now, we need to find the tension in the cord.

At the lowest point, the centripetal force acting on the sphere is equal to the gravitational force. So the tension T in the cord can be found using the following equation:

T = F_c + F_g
T = m*v^2/L + m*g

Plug in the values:

T = 1.5*(10.022)^2/10.9 + 1.5*9.8
T = 13.7523 + 14.7

T = 28.4523 N

Thus, the tension in the cord is approximately 28.4523 N.