Imagine that you swing about your head a ball attached to the end of a string. The ball moves at a constant speed in a horizontal circle.


Part A
If the mass of the ball is 0.160kg and you supply a tension force of 14.5N to the string, what angle would the string make relative to the horizontal?

Sum of Forces in the Y-direction is the following:

T*Sin(theta)=mg

Solve for theta

theta=Sin^-1(mg/T)

Theta=Sin^-1[(0.160kg*9.8m/s^2)/14.5N]

Theta=6.2º

To find the angle that the string makes relative to the horizontal, you can use the concept of forces in circular motion. The tension force in the string provides the centripetal force that keeps the ball moving in a circle.

Step 1: Find the centripetal force acting on the ball.
The centripetal force (F) is given by the equation:
F = m * a

Where:
F is the centripetal force
m is the mass of the ball
a is the centripetal acceleration

Step 2: Find the centripetal acceleration of the ball.
The centripetal acceleration (a) is given by the equation:
a = v^2 / r

Where:
v is the velocity of the ball
r is the radius of the circular path

Step 3: Find the velocity of the ball.
Since the ball moves at a constant speed, the magnitude of the velocity is constant throughout the circular motion. So, the magnitude of the velocity can be directly determined as:
v = sqrt(F / m)

Step 4: Find the angle that the string makes relative to the horizontal.
Let θ be the angle that the string makes with the horizontal. Then, the horizontal component of the tension force (T) can be given as:
T_horizontal = T * cos(θ)

Using trigonometry, we can express the angle θ as:
θ = arccos(T_horizontal / T)

Now, let's calculate the angle θ:

Given:
Mass of the ball (m) = 0.160 kg
Tension force (T) = 14.5 N

Step 1: Find the centripetal force.
F = m * a

Since the ball moves at a constant speed, the acceleration is centripetal acceleration.
F = m * a = m * (v^2 / r)

Step 2: Find the centripetal acceleration.
a = v^2 / r

Step 3: Find the velocity.
v = sqrt(F / m)

Step 4: Find the angle.
θ = arccos(T_horizontal / T)

To find the angle that the string makes with the horizontal, we can use the concept of centripetal force.

The tension force in the string provides the centripetal force required to keep the ball moving in a circle.

First, let's define the variables:
- Tension force (T) = 14.5 N
- Mass of the ball (m) = 0.160 kg
- Angle with the horizontal (θ) = ?

The centripetal force in this case is provided by the tension in the string.
T = Fc (centripetal force)

The centripetal force can be calculated using the formula:
Fc = m * a

The acceleration (a) in circular motion can be found using the formula:
a = v^2 / r

Given that the ball moves at a constant speed, the velocity (v) is constant, and the radius (r) can be considered constant as well.

Now, let's substitute the values into the equations:
Fc = m * a
T = m * (v^2 / r)

Since we are looking for the angle (θ) that the string makes with the horizontal, we can use trigonometric functions.

The horizontal component of the tension force is T * cos(θ).
So, T * cos(θ) = m * (v^2 / r)

Rearranging the equation, we can solve for cos(θ):
cos(θ) = (m * (v^2 / r)) / T

Now, let's substitute the given values:
m = 0.160 kg
T = 14.5 N

Since the constant speed is given, we can assume the radius and velocity are known.

After calculating (v^2 / r) and substituting the values, we can find cos(θ) using the equation. Finally, we can take the inverse cosine (cos^-1) to find the angle θ.