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.
To determine the centripetal acceleration of the bob at the bottom of the arc, we can use the formula:
a = (v^2) / r
Where:
a = centripetal acceleration
v = velocity of the bob
r = radius of the circle (length of the string)
Given:
Length of the string (r) = 1.70 m
Velocity of the bob (v) = 1.50 m/s
Plugging in the values into the formula:
a = (1.50^2) / 1.70
a = 2.25 / 1.70
a = 1.3235 m/s^2
Therefore, the centripetal acceleration of the bob at the bottom of the arc is approximately 1.3235 m/s^2.
To determine the tension in the string at the bottom of the arc, we can use the following equation:
Tension = mg + ma
Where:
Tension = tension in the string
m = mass of the bob
g = acceleration due to gravity (approximately 9.8 m/s^2)
a = centripetal acceleration (which we found to be 1.3235 m/s^2)
Given:
Mass of the bob (m) = 0.490 kg
Plugging in the values:
Tension = (0.490 kg)(9.8 m/s^2) + (0.490 kg)(1.3235 m/s^2)
Tension = 4.802 kg*m/s^2 + 0.648 kg*m/s^2
Tension = 5.45 kg*m/s^2
Therefore, the tension in the string at the bottom of the arc is approximately 5.45 kg*m/s^2.
To determine the centripetal acceleration of the bob at the bottom of the arc, we can use the formula for centripetal acceleration:
a = rω²
where "a" is the centripetal acceleration, "r" is the radius (length of the string), and "ω" is the angular velocity.
To find the angular velocity, we need to consider the initial conditions of the pendulum. Since the pendulum starts out at an angle of 45 degrees below the horizontal and is given an initial downward speed of 1.50 m/s, we can determine the angular velocity using the conservation of mechanical energy.
At the bottom of the arc, all of the initial potential energy is converted into kinetic energy. Therefore, we can equate the initial potential energy to the final kinetic energy:
mgh = (1/2)mv²
where "m" is the mass of the bob, "g" is the acceleration due to gravity, "h" is the height of the pendulum, and "v" is the final velocity.
Given the mass of the bob (0.490 kg) and the height of the pendulum (0.356 m), we can solve for the final velocity:
0.490 kg * 9.8 m/s² * 0.356 m = (1/2) * 0.490 kg * v²
Solving for "v²", we find:
v² = (0.490 kg * 9.8 m/s² * 0.356 m) / (0.490 kg * (1/2))
v² ≈ 3.5176
v ≈ √3.5176 ≈ 1.8754 m/s
Now, we can use the final velocity to find the angular velocity. The relationship between linear velocity (v) and angular velocity (ω) in a rotational motion is:
v = rω
where "r" is the radius (length of the string).
Given the length of the string (1.70 m) and the final velocity (1.8754 m/s), we can rearrange the equation to solve for ω:
1.8754 m/s = 1.70 m * ω
ω = 1.8754 m/s / 1.70 m ≈ 1.1026 rad/s
Finally, with the length of the string (radius) and the angular velocity, we can find the centripetal acceleration:
a = rω²
a = 1.70 m * (1.1026 rad/s)²
a ≈ 2.3853 m/s²
Therefore, the centripetal acceleration of the bob at the bottom of the arc is approximately 2.3853 m/s².
Now, to determine the tension in the string at the bottom of the arc, we can use the equation for net force:
ΣF = T - mg = ma
where "T" is the tension in the string, "m" is the mass of the bob, "g" is the acceleration due to gravity, and "a" is the centripetal acceleration.
Rearranging the equation, we can solve for the tension:
T = mg + ma
T = m(g + a)
Plugging in the values, we get:
T = (0.490 kg)(9.8 m/s² + 2.3853 m/s²)
T ≈ 6.0447 N
Therefore, the tension in the string at the bottom of the arc is approximately 6.0447 N.