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A 16 kg dog is running forward at [tex]$3.20 \, \text{m/s}$[/tex] when it slips on ice and begins to slide. The owner pulls the leash back with a force of 95 N at [tex]$21.8^{\circ}$[/tex] above the horizontal.

Given:
- Static friction coefficient: [tex][tex]$\mu_s = 0.190$[/tex][/tex]
- Kinetic friction coefficient: [tex]$\mu_k = 0.1$[/tex]

Once the dog is at rest, determine whether the continued force is sufficient to cause the dog to start sliding backward.

Sagot :

GinnyAnswer
Certainly! Let's break down the problem step-by-step to determine if the owner's pulling force is sufficient to start sliding the dog backward from rest on ice.

### Step 1: Convert the Dog's Weight to Kilograms

The dog’s weight is given in hectograms (hg):
[tex]\[ 16 \, \text{hg} \][/tex]

We need to convert this to kilograms (kg):
[tex]\[ \text{Weight in kg} = 16 \, \text{hg} \times 0.1 \, \left(\frac{\text{kg}}{\text{hg}}\right) = 1.6 \, \text{kg} \][/tex]

### Step 2: Calculate the Normal Force

The normal force is the perpendicular force exerted by the surface on the dog. This can be calculated using the dog’s weight and the acceleration due to gravity ([tex]\(9.8 \, \text{m/s}^2\)[/tex]):
[tex]\[ \text{Normal Force} = 1.6 \, \text{kg} \times 9.8 \, \frac{\text{m}}{\text{s}^2} = 15.68 \, \text{N} \][/tex]

### Step 3: Calculate the Parallel Component of the Pulling Force

The owner pulls the leash with a force of [tex]\( 95 \, \text{N} \)[/tex] at an angle of [tex]\( 21.8^\circ \)[/tex] above the horizontal. We need to find the component of this force parallel to the ice surface.

First, convert the angle to radians for calculation purposes:
[tex]\[ \text{Angle in radians} = 21.8^\circ \times \left( \frac{\pi}{180} \right) \approx 0.38 \, \text{radians} \][/tex]

Now, calculate the parallel component:
[tex]\[ \text{Parallel Force} = 95 \, \text{N} \times \cos(0.38) \approx 88.21 \, \text{N} \][/tex]

### Step 4: Calculate the Static Frictional Force

The static frictional force can be determined using the coefficient of static friction ([tex]\( \mu_s = 0.19 \)[/tex]) and the normal force:
[tex]\[ \text{Static Frictional Force} = \mu_s \times \text{Normal Force} = 0.19 \times 15.68 \, \text{N} \approx 2.98 \, \text{N} \][/tex]

### Step 5: Determine if the Dog Starts Sliding

To determine if the dog will start sliding, compare the parallel component of the pulling force to the static frictional force. If the parallel force is greater than the static frictional force, the dog will start sliding.

[tex]\[ \text{Parallel Force} = 88.21 \, \text{N} \][/tex]
[tex]\[ \text{Static Frictional Force} = 2.98 \, \text{N} \][/tex]

Since [tex]\( 88.21 \, \text{N} > 2.98 \, \text{N} \)[/tex], the dog's owner’s pulling force is indeed sufficient to overcome the static friction and start sliding the dog backward from rest.

### Conclusion

Hence, the pulling force applied by the owner is strong enough to start sliding the dog backward on the ice from rest.
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