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 45^\circ below the horizontal and is given an initial downward speed of 1.50 m/s.

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.

See my later response to the same question.

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To find the centripetal acceleration of the bob at the bottom of the arc, we can use the formula for centripetal acceleration:

a_c = (v^2) / r

Where:
a_c is the centripetal acceleration
v is the velocity of the bob
r is the radius of the circular path

1. Determine the velocity of the bob at the bottom of the arc:
We are given the initial downward speed of the pendulum, which is 1.50 m/s. As the pendulum swings, the potential energy is converted into kinetic energy, so from the principle of conservation of energy, we know that the kinetic energy at the bottom of the arc is equal to the initial potential energy. The initial potential energy is given by:

PE = m * g * h

Where:
m is the mass of the bob (0.490 kg)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the height of the bob above the lowest point of the arc (since it starts at 45 degrees below the horizontal, the height is 0.17 m * sin(45 degrees))

Substituting the values into the equation, we get:
PE = 0.490 kg * 9.8 m/s^2 * 0.17 m * sin(45 degrees)

2. Calculate the velocity by equating the initial potential energy to the final kinetic energy:
KE = (1/2) * m * v^2

Substituting the values into the equation, we get:
0.490 kg * 9.8 m/s^2 * 0.17 m * sin(45 degrees) = (1/2) * 0.490 kg * v^2

Solving for v, we find:
v = sqrt((9.8 m/s^2 * 0.17 m * sin(45 degrees)) / 2)

Once we have the velocity (v), we can proceed to calculate the centripetal acceleration using the formula mentioned earlier:

a_c = (v^2) / r

Substituting the values of v and r (radius of the circular path = 0.17 m), we can calculate the centripetal acceleration.

To find the tension in the string at the bottom of the arc, we need to consider the two forces acting on the bob: tension force (T) and gravitational force (mg).

At the bottom of the arc, the tension force provides the centripetal force required to keep the bob moving in a circular path. Therefore, the tension force (T) is equal to the net force acting on the bob.

By applying Newton's second law of motion, we can write:

T - mg = m * a_c

Where:
T is the tension in the string
m is the mass of the bob (0.490 kg)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
a_c is the centripetal acceleration

Substituting the values into the equation, we can solve for T to determine the tension in the string at the bottom of the arc.