A skier goes down a slope inclined at 25 degrees. Her mass including equipment is 60.0kg.
a. What is her acceleration if friction is negligible?
b. What is her acceleration if friction is 45.0 N?
netforcedownhill= ma
mgSinTheta-frictionforce=ma
a) 4.1 ms-2
b) 3.4 ms-2
a. To calculate the skier's acceleration when friction is negligible, we can use the formula:
acceleration = g * sin(theta)
where:
acceleration is the skier's acceleration
g is the acceleration due to gravity (approximately 9.8 m/s^2)
theta is the angle of inclination of the slope (25 degrees)
Let's substitute the values into the formula:
acceleration = 9.8 m/s^2 * sin(25 degrees)
Using a calculator, we find:
acceleration ≈ 4.14 m/s^2
Therefore, the skier's acceleration is approximately 4.14 m/s^2 when friction is negligible.
b. To calculate the skier's acceleration when friction is 45.0 N, we can use the formula:
acceleration = (g * sin(theta)) - (friction / mass)
where:
acceleration is the skier's acceleration
g is the acceleration due to gravity (approximately 9.8 m/s^2)
theta is the angle of inclination of the slope (25 degrees)
friction is the magnitude of the frictional force (45.0 N)
mass is the skier's mass including equipment (60.0 kg)
Let's substitute the values into the formula:
acceleration = (9.8 m/s^2 * sin(25 degrees)) - (45.0 N / 60.0 kg)
Using a calculator, we find:
acceleration ≈ 3.68 m/s^2
Therefore, the skier's acceleration is approximately 3.68 m/s^2 when friction is 45.0 N.
To find the skier's acceleration in both cases, we can apply Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
a. If friction is negligible, only the component of the gravitational force parallel to the slope will contribute to the skier's acceleration. We can find this component by multiplying the gravitational force (weight) by the sine of the angle of inclination.
1. Calculate the force due to gravity:
F_gravity = mass * gravity
where mass = 60.0 kg and gravity = 9.8 m/s^2.
2. Calculate the component of the gravitational force parallel to the slope:
F_parallel = F_gravity * sin(angle)
where angle = 25 degrees.
3. Calculate the acceleration:
acceleration = F_parallel / mass
b. If friction is 45.0 N, we need to consider the additional force opposing the motion. This force is subtracted from the component of the gravitational force parallel to the slope.
1. Calculate the net force acting on the skier:
F_net = F_parallel - friction
where friction = 45.0 N.
2. Calculate the acceleration:
acceleration = F_net / mass
Remember to convert the angle to radians if necessary before using trigonometric functions.