Starting from rest, a 10kg suitcase slides down a 3.00m long ramp inclined at 30 degrees from the floor. The coefficient of friction between the suitcase and the ramp is 0.35.
1) what net force is applied to the suitcase while it is on the ramp?
2)what is the change in the kinetic energy of the suitcase as it slides down the ramp?
3) how fast is it traveling when it reaches the bottom of the ramp?
Well, let's see how far down it went:
sin 30 = h/3
so
h = 1.5 meters high
so at the top it has potential energy = m g h = 10*9.81*1.5 = 147 Joules compared to when it hits the ground.
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Let's look at the friction:
weight = m g = 98.1 N
so
normal force on ramp = 98.1 cos 30
= 85 N
so the friction force = .35*85
= 29.7 N
The component of weight down the ramp is
98.1 sin 30 = 49.1 N
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The net force down the ramp is thus
49.1 - 29.7 = 19.4 N (part (1))
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The Ke at bottom is the loss in potential energy - work done by friction
= 147 Joules - 29.7*3 = 147-89.1
= 57.9 Joules
Alternately the work done by the net force down the ramp is
19.4 N
so the work done is 19.4*3 = 58.2 Joules
so call it 58 Joules
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(1/2) m v^2 = 58
v^2 = 58 * 2 /10
v = 3.4 m/s
Well, well, well! Looks like we have ourselves a sliding suitcase situation here. Let's put on our physics hats and solve these questions with a touch of humor, shall we?
1) What net force is applied to the suitcase while it is on the ramp?
To figure this out, we need to consider the forces acting on the suitcase. We have gravitational force pulling it down the ramp, and the frictional force that opposes its motion. The net force is the vector sum of these forces. Now, if only suitcases came with tiny seat belts! Imagine how much fun that would be!
Anyway, enough clowning around. The gravitational force acting on the suitcase can be calculated using the equation Fg = m * g. Plugging in the values, we get:
Fg = 10kg * 9.8 m/s² = 98 N (Newtons)
Now, the frictional force can be calculated using the equation Ff = μ * Fn, where μ is the coefficient of friction and Fn is the normal force. But we need to find the normal force first! So, we need to use some trigonometry to break down the gravitational force into components.
Fn = Fg * cos(θ), where θ is the angle of inclination. Plugging in the values, we get:
Fn = 98 N * cos(30°) = 84.85 N (approximately)
Finally, we can calculate the frictional force:
Ff = μ * Fn = 0.35 * 84.85 N = 29.69 N (approximately)
Now, to calculate the net force, we subtract the frictional force from the gravitational force:
Net force = Fg - Ff = 98 N - 29.69 N = 68.31 N (approximately)
So, the net force applied to the suitcase on the ramp is approximately 68.31 Newtons. It's like a mini wrestling match between gravity and friction!
2) What is the change in the kinetic energy of the suitcase as it slides down the ramp?
The change in kinetic energy is simply the difference between the final and initial kinetic energies. Since the suitcase starts from rest, we only need to calculate the final kinetic energy. Are you ready?
The final kinetic energy can be calculated using the equation Kf = (1/2) * m * v², where m is the mass and v is the final velocity. We can use the following equation to find the final velocity:
vf² = vi² + 2 * a * d
Considering the suitcase is not in free fall, but rather sliding, we need to calculate the acceleration using the net force:
a = Fnet / m = 68.31 N / 10 kg = 6.831 m/s² (approximately)
Now we can find the final velocity:
vf² = 0² + 2 * 6.831 m/s² * 3.00 m
vf² = 41.01 m²/s²
vf ≈ √41.01 m²/s² ≈ 6.41 m/s (approximately)
Finally, we can calculate the change in kinetic energy:
ΔK = Kf - Ki = (1/2) * m * vf² - (1/2) * m * 0²
ΔK ≈ (1/2) * 10 kg * (6.41 m/s)² - 0
ΔK ≈ 128.3 J (approximately)
Voila! The change in the kinetic energy of the suitcase as it slides down the ramp is approximately 128.3 Joules. That's like a rollercoaster ride for your clothing!
3) How fast is it traveling when it reaches the bottom of the ramp?
Ah, the grand finale! To find the speed at the bottom, we take a deep breath and dive into the equations again. Ready? Here we go!
We know the final velocity vf from the previous question: approximately 6.41 m/s. So, it's like the suitcase has had a little caffeine boost after sliding down the ramp!
Now all that's left is to enjoy some simple trigonometry. We can find the horizontal component of the velocity using:
Vx = vf * cos(θ)
Plugging in the values:
Vx = 6.41 m/s * cos(30°)
Vx ≈ 5.54 m/s (approximately)
Ta-da! The suitcase is traveling horizontally at a whopping speed of approximately 5.54 meters per second when it reaches the bottom of the ramp. It's almost like a mini-superhero taking flight!
Remember, this answer is all just a clowning approximation, so don't forget to take the real world into account!
To solve these questions, we need to break them down step-by-step. Let's begin!
1) Calculating the Net Force:
To find the net force acting on the suitcase, we need to consider the forces acting on it.
The forces acting on the suitcase on the ramp are:
- The weight of the suitcase (mg), which is acting vertically downward.
- The normal force (N), which acts perpendicular to the surface of the ramp.
- The force of friction (f), which opposes the motion of the suitcase.
The weight of the suitcase (mg) can be calculated as:
Weight = mass x gravity
Weight = 10kg x 9.8 m/s^2 (acceleration due to gravity)
Weight = 98N
The normal force (N) can be calculated as:
Normal Force = Weight x cos(θ)
Normal Force = 98N x cos(30°)
Normal Force = 98N x 0.866
Normal Force = 84.94N
The force of friction (f) can be calculated as:
Force of Friction = coefficient of friction x Normal Force
Force of Friction = 0.35 x 84.94N
Force of Friction = 29.73N
The net force (F_net) can be calculated as:
Net Force = Weight - Force of Friction
Net Force = 98N - 29.73N
Net Force = 68.27N
Therefore, the net force applied to the suitcase while it is on the ramp is 68.27N.
2) Calculating the Change in Kinetic Energy:
The change in kinetic energy can be calculated using the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.
The work done on the suitcase is equal to the net force applied to it multiplied by the distance traveled along the ramp:
Work = Net Force x Distance
Work = 68.27N x 3.00m
Work = 204.81J (Joules)
Since the suitcase starts from rest, its initial kinetic energy is zero.
The change in kinetic energy (ΔKE) can be calculated as:
ΔKE = Work
Therefore, the change in kinetic energy of the suitcase as it slides down the ramp is 204.81J.
3) Calculating the Final Velocity:
To find the speed of the suitcase when it reaches the bottom of the ramp, we can use the work-energy principle again.
The work done on the suitcase is equal to the change in its kinetic energy:
Work = ΔKE
The work done can also be calculated as the force of gravity (Weight) multiplied by the vertical distance (h) the suitcase travels:
Work = Weight x h
h can be calculated using the equation:
h = Distance x sin(θ)
h = 3.00m x sin(30°)
h = 1.50m
Therefore, we have:
Work = Weight x h
204.81J = 98N x 1.50m
Now, we can calculate the final velocity (v) using the kinetic energy formula:
ΔKE = 1/2mv^2
v = √(2ΔKE/m)
v = √(2 x 204.81J / 10kg)
v ≈ 9.05 m/s
Therefore, the suitcase is traveling at approximately 9.05 m/s when it reaches the bottom of the ramp.
To find the answers to these questions, we need to go through a series of steps using Newton's laws of motion. Let's break down the problem and find the solutions step by step.
Step 1: Determine the gravitational force.
The first step is to calculate the gravitational force acting on the suitcase. The gravitational force can be calculated using Newton's second law formula: F = m * g, where F is the force, m is the mass of the suitcase (10 kg in this case), and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_grav = 10 kg * 9.8 m/s^2 = 98 N
Step 2: Find the force component parallel to the ramp.
The gravitational force will be split into two components: one parallel to the ramp and one perpendicular to the ramp. To find the component parallel to the ramp, we use the formula: F_parallel = F_grav * sin(theta), where theta is the angle of the ramp (30 degrees).
F_parallel = 98 N * sin(30) = 49 N
Step 3: Calculate the force of friction.
The force of friction can be determined using the equation: F_friction = coefficient_of_friction * F_normal, where coefficient_of_friction is given as 0.35 and F_normal is the normal force acting on the suitcase. The normal force is equal to the component of the gravitational force perpendicular to the ramp, which can be calculated as: F_normal = F_grav * cos(theta).
F_normal = 98 N * cos(30) = 84.852 N
F_friction = 0.35 * 84.852 N = 29.598 N
Step 4: Calculate the net force.
The net force acting on the suitcase is the difference between the parallel component of the gravitational force and the force of friction. Since both these forces act in opposite directions along the ramp, the net force can be found using the formula: F_net = F_parallel - F_friction.
F_net = 49 N - 29.598 N = 19.402 N
So, the net force applied to the suitcase while it is on the ramp is 19.402 N.
Step 5: Calculate the change in kinetic energy.
The change in kinetic energy of the suitcase can be found using the formula: ΔKE = (1/2) * m * v^2, where ΔKE is the change in kinetic energy, m is the mass of the suitcase, and v is the final velocity of the suitcase.
We can assume that the suitcase starts from rest, so the initial velocity (u) is 0. The final velocity (v) can be calculated using the formula: v = sqrt(2 * a * s), where a is the acceleration and s is the distance traveled along the ramp.
The acceleration (a) along the ramp can be found using the formula: a = F_net / m. Plugging in the values, we get:
a = 19.402 N / 10 kg = 1.9402 m/s^2
The distance (s) traveled along the ramp is given as 3.00 m.
v = sqrt(2 * 1.9402 m/s^2 * 3.00 m) = 3.735 m/s
Now, we can use this final velocity to calculate the change in kinetic energy:
ΔKE = (1/2) * 10 kg * (3.735 m/s)^2 = 69.924 J
So, the change in kinetic energy of the suitcase as it slides down the ramp is 69.924 joules.
Step 6: Calculate the final velocity.
Finally, we need to find the speed (or final velocity) at the bottom of the ramp. We can use the same formula used in step 5 to find the final velocity (v) when s = 3.00 m:
v = sqrt(2 * 1.9402 m/s^2 * 3.00 m) = 3.735 m/s
Therefore, when the suitcase reaches the bottom of the ramp, it will be traveling at 3.735 m/s.