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.

Where is "the horizontal" and what does ^\circ mean?

This looks like a conservation of energy problem.

To determine 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 at the bottom of the arc, and
r is the radius of the circular motion, which is equal to the length of the string.

In this case, the length of the string is given as 1.70 m. To find the velocity of the bob at the bottom of the arc, we need to use the conservation of mechanical energy:

E_m = E_k + E_p

where
E_m is the total mechanical energy of the system,
E_k is the kinetic energy of the bob, and
E_p is the potential energy of the bob at the given position.

At the bottom of the arc, the bob is at its lowest point, so its potential energy is at its minimum and kinetic energy is at its maximum.

According to the conservation of mechanical energy, the initial mechanical energy is equal to the energy at the bottom of the arc:

E_m_initial = E_m_bottom

E_k_initial + E_p_initial = E_k_bottom + E_p_bottom

In this case, the initial energy is given by the initial downward speed:

E_m_initial = (1/2) * m * v_initial^2

where
m is the mass of the bob, given as 0.490 kg, and
v_initial is the initial downward speed, given as 1.50 m/s.

At the bottom of the arc, the kinetic energy is maximum and the potential energy is minimum. The potential energy at the bottom of the arc is zero since the bob is at its lowest position.

E_k_bottom = (1/2) * m * v_bottom^2

E_p_bottom = 0

Using these equations, we can solve for v_bottom:

E_m_initial = E_k_bottom + E_p_bottom

(1/2) * m * v_initial^2 = (1/2) * m * v_bottom^2

Simplifying the equation, we get:

v_bottom = sqrt(v_initial^2)

Now we can substitute the calculated value of v_bottom and the given value of the radius r into the formula for centripetal acceleration:

a_c = (v_bottom^2) / r

Once we have the value of centripetal acceleration, we can calculate the tension in the string at the bottom of the arc.

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

At the bottom of the arc, the tension in the string provides the necessary centripetal force to keep the bob in circular motion. The gravitational force acts vertically downward.

Since the bob is at its lowest point and the string is making an angle of 45^\circ below the horizontal, we can use trigonometry to find the tension in the string.

The vertical component of the tension (T_y) should be equal to the gravitational force (mg). Using trigonometry, we can write:

T_y = mg * cos(45^\circ)

Lastly, we can use the Pythagorean theorem to find the tension in the string (T):

T = sqrt(T_y^2 + T_x^2)

where
T_x is the horizontal component of the tension, which can be found using the trigonometric relationship:

T_x = T * sin(45^\circ)