A 59-kg person on skis is going down a hill sloped at 30° from the horizontal. The coefficient of friction between the skis and the snow is 0.10. What would be the magnitude of the acceleration?
net force= mass*acceleration
mg*sinTheta-mg*mu*cosTheta=mass*a
solvefor a. Notice mass divides out.
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To find the magnitude of the acceleration, we can break down the forces acting on the person on skis:
1. Force due to gravity:
The force due to gravity can be calculated using the equation F = mg, where m is the mass (59 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2). Thus, the force due to gravity is Fg = (59 kg)(9.8 m/s^2) = 578.2 N.
2. Force of friction:
The force of friction can be calculated using the equation Ff = μN, where μ is the coefficient of friction (0.10) and N is the normal force. The normal force is equal to the component of the force due to gravity perpendicular to the surface, which can be calculated as N = mgcosθ, where θ is the angle of the slope (30°). Thus, the normal force is N = (59 kg)(9.8 m/s^2)cos(30°) = 509.00 N. Therefore, the force of friction is Ff = (0.10)(509.00 N) = 50.90 N.
3. Net force:
The net force can be calculated as the vector sum of the force due to gravity and the force of friction. Since the force due to gravity is acting directly down the slope and the force of friction acts in the opposite direction, they are in the opposite direction and can be subtracted. Thus, the net force is Fnet = Fg - Ff = 578.2 N - 50.90 N = 527.30 N.
4. Magnitude of acceleration:
The magnitude of the acceleration can be determined using Newton's second law, Fnet = ma, where Fnet is the net force and m is the mass. Rearranging the equation, we find a = Fnet/m. Substituting the values, we get a = (527.30 N) / (59 kg) = 8.93 m/s^2.
Therefore, the magnitude of the acceleration of the person on skis going down the hill is approximately 8.93 m/s^2.
To find the magnitude of the acceleration, we need to consider the forces acting on the person on skis.
First, let's analyze the gravitational force acting on the person. The gravitational force can be calculated using the equation:
F_gravity = mass * acceleration_due_to_gravity
where the mass is given as 59 kg and the acceleration due to gravity is approximately 9.8 m/s^2.
F_gravity = 59 kg * 9.8 m/s^2 = 578.2 N
Next, we need to find the force of friction acting on the skis. The force of friction can be calculated using the equation:
F_friction = coefficient_of_friction * F_normal
where the coefficient of friction is given as 0.10.
The normal force, F_normal, is equal to the gravitational force acting perpendicular to the slope. In this case, the normal force can be calculated as:
F_normal = mass * acceleration_due_to_gravity * cos(theta)
where theta is the angle of the slope, given as 30 degrees.
F_normal = 59 kg * 9.8 m/s^2 * cos(30°) = 509.7 N
Now, we can find the force of friction:
F_friction = 0.10 * 509.7 N = 50.97 N
The net force, F_net, acting on the person can be calculated by subtracting the force of friction from the gravitational force:
F_net = F_gravity - F_friction = 578.2 N - 50.97 N = 527.23 N
Finally, we can calculate the acceleration using the equation:
F_net = mass * acceleration
Therefore,
acceleration = F_net / mass = 527.23 N / 59 kg = 8.92 m/s^2
So, the magnitude of the acceleration is 8.92 m/s^2.