A sled and rider have a total mass of 57.2 kg. They are on a snowy hill accelerating at 0.5g. The coefficient of kinetic friction between the sled and the snow is 0.20. What is the angle of the hill's slope measured upward from the horizontal? You may find a spreadsheet program helpful in answering this question.
Harry Potter decides to take Pottery 101 as an elective to satisfy his arts requirement at Hogwarts. He sets some clay
(m = 4.00 kg)
on the edge of a pottery wheel
(r = 0.730 m),
which is initially motionless. He then begins to rotate the wheel with a uniform acceleration, reaching a final angular speed of 2.800 rev/s in 2.80 s.
(a) What is the speed of the clay when the initial 2.80 s has passed?
m/s
(b) What is the centripetal acceleration of the clay initially and when the initial 2.80 s has passed? (Enter the magnitudes of your answers.)
ac,i = m/s^2
ac,f = m/s^2
(c) What is the magnitude of the constant tangential acceleration responsible for starting the clay in circular motion?
m/s^2
To find the angle of the hill's slope, we can use the following steps:
1. Start by considering the forces acting on the sled and rider on the hill. There are two main forces to consider: the gravitational force (mg) acting vertically downward and the frictional force (f) acting opposite to the direction of motion.
2. The equation for the force of friction is given by f = μN, where μ is the coefficient of kinetic friction and N is the normal force. The normal force can be calculated as N = mg - ma, where a is the acceleration of the sled and rider.
3. Since the sled and rider are on a slope, the normal force can be split into two components: Ncosθ acting perpendicular to the slope and Nsinθ acting parallel to the slope, where θ is the angle of the slope measured upward from the horizontal.
4. The gravitational force can also be split into two components: mgcosθ acting perpendicular to the slope and mgsinθ acting parallel to the slope.
5. Using Newton's second law, F = ma, we can write the equation for the forces acting parallel to the slope as follows: mgsinθ - f = ma.
6. Substituting the equation for the force of friction (f = μN) and the equation for the normal force (N = mg - ma), we get: mgsinθ - μ(mg - ma) = ma.
7. Simplify the equation by canceling out common terms and rearranging to solve for the angle θ.
8. This equation involves both mass (m) and acceleration (a). Based on the given information, we know that the sled and rider have a total mass of 57.2 kg and are accelerating at 0.5g. Therefore, we need to substitute these values into the equation.
9. To perform the calculations, using a spreadsheet program such as Microsoft Excel or Google Sheets can be helpful. Set up the equation in the spreadsheet and plug in the known values to find the angle θ.
10. Once you have the equation set up in the spreadsheet, you can use various mathematical functions available in the program to solve for the angle of the slope (θ).
By following these steps and using a spreadsheet program to do the calculations, you can determine the angle of the hill's slope measured upward from the horizontal.