A person, sunbathing on a warm day, is lying horizontally on the deck of a boat. Her mass is 59.6 kg, and the coefficient of static friction between the deck and her is 0.696. Assume that she is moving horizontally, and that the static frictional force is the only force acting on her in this direction. (a) What is the magnitude of the static frictional force when the boat moves with a constant velocity of +5.36 m/s? (b) The boat speeds up with an acceleration of 1.41 m/s2, and she does not slip with respect to the deck. What is the magnitude of the static frictional force that acts on her? (c) What is the magnitude of the maximum acceleration the boat can have before she begins to slip relative to the deck?

a) zero, not accelerating

b) F = m a
= 59.6(1.41)

c) F = m g mu = m a
a = mu g
a = .696 * 9.81
= 6.83 m/s^2

To determine the magnitude of the static frictional force in various scenarios, we can use the following equations:

(a) The static frictional force when the boat moves with a constant velocity can be calculated using the equation:

Static frictional force (Fs) = coefficient of static friction (μs) × normal force (Fn)

Since the person is lying horizontally, the normal force is equal to her weight:

Fn = mass × acceleration due to gravity

Let's plug in the given values:

Mass (m) = 59.6 kg
Coefficient of static friction (μs) = 0.696
Acceleration due to gravity (g) = 9.8 m/s^2

Fn = 59.6 kg × 9.8 m/s^2

Now, we can calculate the static frictional force:

Fs = 0.696 × (59.6 kg × 9.8 m/s^2)

(b) In this scenario, the boat speeds up with an acceleration, and the person does not slip. Therefore, the static frictional force will exactly match the force required to accelerate the person with the boat. The magnitude of the static frictional force can be calculated using the equation:

Fs = mass × acceleration

Given values are as follows:

Mass (m) = 59.6 kg
Acceleration (a) = 1.41 m/s^2

Fs = 59.6 kg × 1.41 m/s^2

(c) To determine the maximum acceleration the boat can have before the person begins to slip, we need to find the point where the static frictional force reaches its maximum value. This occurs when the applied force is equal to the maximum static frictional force. The maximum static frictional force can be calculated using the equation:

Maximum static frictional force (Fmax) = coefficient of static friction (μs) × normal force (Fn)

Using the same normal force as before:

Fn = 59.6 kg × 9.8 m/s^2

Now we can calculate the maximum static frictional force:

Fmax = 0.696 × (59.6 kg × 9.8 m/s^2)

Finally, we can determine the maximum acceleration by dividing the maximum static frictional force by the person's mass:

Maximum acceleration = Fmax / mass

To solve this problem, we will use the concept of static friction and Newton's second law of motion.

First, let's understand the concept of static friction. Static friction is the force that prevents two surfaces from sliding against each other when they are in contact but not moving relative to each other.

Now let's find the magnitude of the static frictional force when the boat moves with a constant velocity of +5.36 m/s (part a).

Since the boat is moving with a constant velocity, the net force acting on the person is zero. Therefore, the magnitude of the static frictional force (F_friction) must be equal in magnitude but opposite in direction to the force applied on the person.

We know that the static frictional force (F_friction) can be calculated using the formula:

F_friction = μ_s * N

where μ_s is the coefficient of static friction and N is the normal force.

In this case, the normal force is the person's weight (mg), which can be calculated using:

N = mg

where m is the mass of the person and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Substituting the given values:

m = 59.6 kg
μ_s = 0.696
g = 9.8 m/s^2

N = (59.6 kg) * (9.8 m/s^2) = 584.08 N

F_friction = (0.696) * (584.08 N) = 406.32 N

Therefore, the magnitude of the static frictional force when the boat moves with a constant velocity of +5.36 m/s is 406.32 N.

Moving on to part b, where the boat speeds up with an acceleration of 1.41 m/s^2 and she does not slip with respect to the deck.

In this case, the person experiences an additional force due to the acceleration of the boat. The static frictional force (F_friction) must be equal in magnitude but opposite in direction to the sum of the force applied on the person and the force due to the acceleration.

Using Newton's second law of motion, F_net = m * a, where F_net is the net force, m is the mass, and a is the acceleration, we can set up the following equation:

F_net = F_applied + F_acceleration + F_friction

Since the boat is moving horizontally, the only force applied on the person is the force due to acceleration, F_applied = m * a.

Substituting the given values:

m = 59.6 kg
a = 1.41 m/s^2

F_applied = (59.6 kg) * (1.41 m/s^2) = 83.836 N

Now, we can rewrite the equation as:

F_friction = F_net - F_applied - F_acceleration

Since F_net is zero (as the person does not slip), the equation simplifies to:

F_friction = -F_applied - F_acceleration

F_friction = -(83.836 N) - F_acceleration

To find the magnitude of the static frictional force, we need to determine the maximum possible value of F_friction, which occurs when the person is at the verge of slipping.

The maximum possible value of static friction (F_friction_max) can be calculated using the formula:

F_friction_max = μ_s * N

Using the same value for N calculated earlier (584.08 N) and the given coefficient of static friction (μ_s = 0.696), we can determine F_friction_max.

F_friction_max = (0.696) * (584.08 N) = 406.32 N

Since the person does not slip, the magnitude of the static frictional force (F_friction) must be less than or equal to F_friction_max.

Therefore, the magnitude of the static frictional force that acts on her when the boat speeds up with an acceleration of 1.41 m/s^2 is 406.32 N.

Moving on to part c, where we need to find the maximum acceleration the boat can have before she begins to slip relative to the deck.

As determined earlier, the maximum possible value of static friction (F_friction_max) is 406.32 N.

Using the same equation for F_friction_max:

F_friction_max = μ_s * N

F_friction_max can also be written as:

F_friction_max = m * a_max

where a_max is the maximum acceleration.

Rearranging the equation:

a_max = F_friction_max / m

a_max = (406.32 N) / (59.6 kg) = 6.826 m/s^2

Therefore, the magnitude of the maximum acceleration the boat can have before she begins to slip relative to the deck is 6.826 m/s^2.