A simple pendulum consists of a light string 1.70 m long with a small 0.490 kg mass attached. The pendulum starts out at 45º below the horizontal and is given an initial downward speed of 1.50 m/s.

find:

At the bottom of the arc, determine the centripetal acceleration of the bob.

At the bottom of the arc, determine the tension in the string.

the KE at the bottom equals the initial PE (mgh) 0f the bob.

1/2 mv^2=mg(1.70cos45)
solve for v. then V^2/r

tension= mg+mv^2/r

To determine the centripetal acceleration of the bob at the bottom of the arc, we can use the equation for centripetal acceleration:

a = (v^2) / r

where:
a = centripetal acceleration
v = velocity of the bob
r = radius or length of the pendulum

First, we need to find the velocity of the bob at the bottom of the arc. We know that the initial speed of the bob was 1.50 m/s and the pendulum is a simple pendulum swinging back and forth. So, at the bottom of the arc, the bob's speed would be maximum:

v = maximum speed = 1.50 m/s

Next, we need to determine the length or radius of the pendulum. In this case, the length of the pendulum is given as 1.70 m.

So, using the given values, we can calculate the centripetal acceleration at the bottom of the arc:

a = (1.50^2) / 1.70
a = 2.25 / 1.70
a = 1.32 m/s^2

Therefore, the centripetal acceleration of the bob at the bottom of the arc is 1.32 m/s^2.

To determine the tension in the string at the bottom of the arc, we can use the equation for centripetal force:

F = m * a

where:
F = tension in the string
m = mass of the bob
a = centripetal acceleration

First, we need to find the mass of the bob. In this case, the mass is given as 0.490 kg.

So, using the given values, we can calculate the tension in the string at the bottom of the arc:

F = (0.490 kg) * (1.32 m/s^2)
F = 0.6468 kg * m/s^2

Therefore, the tension in the string at the bottom of the arc is approximately 0.647 N.