I've been doing alot of extra questions for physics lately, and these four are still giving me trouble. Please help me out and help me solve them, including Free Body Diagrams if possible.
1. If a curve with a radius of 60m is properly banked for a car travelling at 60km/hr, what must the coefficient of static friction be for a car not to skid when travelling at 90km/hr?
2. A satellite of mass "m" moves in a circular orbit about the earth at a height "h" above its surface. If the radius of the earth is R and the accel due to gravity at the earth's surface is "g" show that the period of the satellite can be expressed as T=2pi*route((R+h)^3 / gR^2)
3. Calculate the mass of the sun using the fact that the period of the earth is 3.16x10^7 seconds and its mean distance from the sun is 1.5x10^11m. (G=6.67x10^-11 Nm^2/kg^2)
4. Conical pendulum question-
An amusement park ride consists of a rotating circular platform 8m in diameter from which seats are suspended at the end of 2.5m chains. When the system rotates the cahins holding the seats make an angle theta=28degrees with the vertical a) what is the speed of the seat? b) If a child of mass 40kg sits in the 10kg seat, what is the tension in the chain?
Thanks huge guys, if I can see how these questions are done, i should be set for my midterm exam.
I strongly recommend you obtain a copy of Schaum's Outline Series for Physics, available at any college bookstore, or at Barnes Noble. Very reasonalbly priced, and a valuable asset of solve problems.
Morning bump!
Sure, I can help you solve these problems. Let's start with the first problem:
1. To find the coefficient of static friction, we need to determine the maximum speed at which the car can travel without skidding. We can begin by determining the angle of banking required.
The formula for the angle of banking is given by:
θ = arctan(v^2 / (g * r))
where θ is the angle of banking, v is the velocity of the car, g is the acceleration due to gravity, and r is the radius of the curve.
Plugging in the values, we have:
θ = arctan((60 km/hr)^2 / (9.8 m/s^2 * 60 m))
Next, we can find the coefficient of static friction (μ) using the following equation:
μ = tan(θ)
Now, we need to calculate the maximum speed at which the car can travel without skidding. We can use the following equation:
v_max = sqrt(μ * g * r)
Plugging in the values, we have:
v_max = sqrt(μ * 9.8 m/s^2 * 60 m)
To find the coefficient of static friction for the car not to skid at 90 km/hr, we need to set the maximum speed equal to 90 km/hr and solve for μ:
90 km/hr = sqrt(μ * 9.8 m/s^2 * 60 m)
Solve the above equation to find the coefficient of static friction (μ) for the car.
Now, let's move on to the second problem:
2. The period of a satellite moving in a circular orbit can be determined using the formula:
T = 2π * sqrt((r^3) / (GM))
where T is the period of the satellite, r is the radius of the orbit, G is the gravitational constant, and M is the mass of the planet (in this case, the Earth).
In this problem, the height above the Earth's surface is given as 'h,' so the radius of the orbit can be expressed as (R + h), where R is the radius of the Earth. Therefore, we have:
r = R + h
To express the period of the satellite as a function of R, h, and g, we replace M with the mass of the Earth, which can be written as M = gR^2 / G, where g is the acceleration due to gravity at the Earth's surface.
Substituting the values in the formula T = 2π * sqrt((r^3) / (GM)), where r = R + h and M = gR^2 / G, we can simplify to obtain the final expression for the period of the satellite, T.
Now, let's move on to the third problem:
3. To calculate the mass of the Sun, we can use Kepler's third law of planetary motion. This law states that the square of the period of a planet (T) is directly proportional to the cube of its mean distance from the Sun (r).
We are given the period of the Earth (T = 3.16x10^7 seconds) and its mean distance from the Sun (r = 1.5x10^11 m). We can set up the following equation:
(T^2) = k * (r^3)
where k is the constant of proportionality.
By rearranging the equation, we have:
k = (T^2) / (r^3)
Now, we need to substitute the values for T and r into the equation to find the value of k. Once we have the value of k, we can calculate the mass of the Sun using the formula:
mass of the Sun = (4π^2 * r^3) / (GT^2)
where G is the gravitational constant.
Now, let's move on to the fourth problem:
4. For the conical pendulum problem, we can start with a free body diagram to analyze the forces acting on the system.
a) The speed of the seat can be determined using the formula for the speed of an object in circular motion. The speed (v) is given by the equation:
v = √(g * L * tanθ)
where g is the acceleration due to gravity, L is the length of the chain, and θ is the angle made by the chain with the vertical.
Substituting the given values of L (2.5 m) and θ (28 degrees) into the equation, we can calculate the speed (v) of the seat.
b) To find the tension in the chain, we need to consider the forces acting on the child. The tension force in the chain (T) and the force due to gravity (mg) are the two forces in equilibrium.
Using the equation for the horizontal component of the tension force (T_hor) in a conical pendulum:
T_hor = T * cosθ
We can equate T_hor to the centripetal force (mv^2 / r), which is the force needed to keep the child moving in a circular path:
T_hor = (m * v^2) / r
Substituting the given values of m (40 kg), v (calculated in part a), and r (4 m, which is half the diameter of the rotating platform), we can solve for the tension force (T).
I hope this helps you understand how to approach and solve these physics problems.