A mass of 8.10 kg is suspended from a 1.69 m long string. It revolves in a horizontal circle as shown in the figure.
The tangential speed of the mass is 2.92 m/s. Calculate the angle between the string and the vertical.
To find the angle between the string and the vertical, we can use the concept of centripetal force.
The centripetal force acting on an object moving in a circle can be calculated using the equation:
Fc = (mv^2) / r
Where:
- Fc is the centripetal force
- m is the mass of the object
- v is the tangential speed of the object
- r is the radius of the circular motion
We can rearrange this equation to solve for the radius:
r = (mv^2) / Fc
Given:
- mass (m) = 8.10 kg
- tangential speed (v) = 2.92 m/s
- radius (r) = 1.69 m
First, let's calculate the centripetal force:
Fc = (8.10 kg)(2.92 m/s)^2 / (1.69 m)
Fc = 36.27837 N (rounded to 3 decimal places)
Now, let's determine the angle between the string and the vertical. In this case, the weight of the mass is acting vertically downwards and is balanced by the tension in the string, which acts along the string.
The weight of the mass (mg) can be calculated using the equation:
mg = mass (m) x acceleration due to gravity (g)
Where:
- g is the acceleration due to gravity, which is approximately 9.8 m/s^2
mg = (8.10 kg)(9.8 m/s^2)
mg = 79.38 N
Since the tension in the string balances the weight of the mass, the tension equals the weight:
Tension = Weight = 79.38 N
Now, let's draw a diagram to visualize the forces acting on the mass:
/|
/ |
/ |
/ θ |
/____| <-- Tension in the string
mg
The vertical component of the tension force can be obtained using trigonometry:
Tension_vertical = Tension * cos(θ)
The weight (mg) can be resolved into its vertical component:
Weight_vertical = mg * sin(θ)
Since the tension in the string and the weight balance each other, we can set the equations for the vertical components of tension and weight equal to each other:
Tension_vertical = Weight_vertical
Tension * cos(θ) = mg * sin(θ)
Substituting the values we have:
79.38 N * cos(θ) = (8.10 kg)(9.8 m/s^2) * sin(θ)
Now, we can solve for θ by rearranging the equation:
cos(θ) / sin(θ) = (8.10 kg)(9.8 m/s^2) / 79.38 N
tan(θ) = (8.10 kg)(9.8 m/s^2) / 79.38 N
θ = arctan[(8.10 kg)(9.8 m/s^2) / 79.38 N]
Using a calculator, we can find:
θ ≈ 71.84 degrees
Therefore, the angle between the string and the vertical is approximately 71.84 degrees.
To calculate the angle between the string and the vertical, we can use the concept of centripetal force.
The centripetal force is the force required to keep an object moving in a circular path. In this case, the centripetal force is provided by the tension in the string.
We can start by calculating the tension in the string. The tension is equal to the centripetal force, which can be calculated using the following formula:
F = m * v^2 / r
where F is the centripetal force, m is the mass of the object, v is the tangential speed, and r is the radius (in this case, the length of the string).
Plugging in the values given:
F = (8.10 kg) * (2.92 m/s)^2 / 1.69 m
F = 34.7822 N
Now, we need to consider the forces acting on the object. There are two forces: the tension in the string and the weight of the object.
The weight of the object can be calculated using the formula:
W = m * g
where W is the weight, m is the mass, and g is the acceleration due to gravity.
Plugging in the values:
W = (8.10 kg) * 9.8 m/s^2
W = 79.38 N
The vertical component of the tension is equal to the weight of the object since they balance each other out.
Now, we can calculate the angle between the string and the vertical. We can use the trigonometric relationship of the forces:
Tension = Weight / cos(theta)
Solving for theta:
cos(theta) = Weight / Tension
cos(theta) = 79.38 N / 34.7822 N
theta = cos^-1(79.38 N / 34.7822 N)
Using a calculator, we find:
theta ≈ 63.8 degrees
So, the angle between the string and the vertical is approximately 63.8 degrees.
The method of solution can be found at
http://en.wikipedia.org/wiki/Conical_pendulum