A 18.5kg penguin slides at a constant velocity of 1.40 m/s down an icy incline. The incline slopes above the horizontal at an angle of 6.90 degrees. At the bottom of the incline the penguin slides onto a horizontal patch of ice. The coefficient of kinetic friction between the penguin and the ice is the same for the incline as for the horizontal patch. During the time the penguin is on the horizontal patch what is the normal force on the penguin? the friction force on the penguin? the acceleration for the penguin? the amount of time required for the penguin to slide to a halt after entering the horizontal patch of ice?
To solve this problem, we can break it down into different stages and analyze each part separately.
1. On the incline:
The penguin is sliding down the incline at a constant velocity. This means that the net force on the penguin is zero. The forces acting on the penguin on the incline are gravity (mg) and the force of friction (μN), where μ is the coefficient of kinetic friction and N is the normal force.
The force of friction can be calculated using the equation: friction force = μN.
Since the penguin is moving at a constant velocity, the force of friction must be equal in magnitude and opposite in direction to the component of the gravitational force parallel to the incline.
Thus, the force of friction is equal to the component of the gravitational force parallel to the incline:
friction force = mg*sin(6.90 degrees).
The normal force can be calculated using the equation: normal force = mg*cos(6.90 degrees).
2. On the horizontal patch:
Once the penguin reaches the horizontal patch, it experiences the same coefficient of kinetic friction. The net force acting on the penguin is now the force of friction.
The acceleration of the penguin can be calculated using Newton's second law: net force = mass * acceleration.
Since the net force is equal to the force of friction, we can write: force of friction = mass * acceleration.
We can substitute the expression for the force of friction from the incline stage (mg*sin(6.90 degrees)) into the equation above.
3. Time required to slide to a halt:
To find the amount of time required for the penguin to slide to a halt, we can use the equation: final velocity = initial velocity + acceleration * time.
The initial velocity is 1.40 m/s (given), the final velocity is 0 m/s (since the penguin comes to a halt), and the acceleration is calculated in step 2.
Useful formulas:
- Force of friction = μN
- Normal force = mg*cos(6.90 degrees)
- Net force = mass * acceleration
- Final velocity = initial velocity + acceleration * time
Now let's calculate each part:
1. Normal force on the penguin:
Normal force = mg*cos(6.90 degrees) = 18.5 kg * 9.8 m/s^2 * cos(6.90 degrees)
2. Friction force on the penguin on the horizontal patch:
Friction force = mg*sin(6.90 degrees) = 18.5 kg * 9.8 m/s^2 * sin(6.90 degrees)
3. Acceleration for the penguin on the horizontal patch:
Force of friction = mass * acceleration
mg*sin(6.90 degrees) = 18.5 kg * acceleration
Acceleration = mg*sin(6.90 degrees) / 18.5 kg
4. Time required for the penguin to slide to a halt:
Final velocity = initial velocity + acceleration * time
0 m/s = 1.40 m/s + (mg*sin(6.90 degrees) / 18.5 kg) * time
Solve for time.