Starting from rest, a 10.0 Kg suitcase slides down a 3.00m long ramp inclined at 30.0 degrees from the floor. The coefficient of friction between the suitcase and ramp is .35. A: what net force is applied to suitcase while it is on the ramp? B: what is the change in the kinetic energy of the suitcase as it slides down the ramp? C: how fast is it traveling when it reaches the bottom of the ramp?
Net force down=gravity-friction
= mg*sinTheta-mg(mu)cosTheta
changeKE=finalKE-initial KE
= netforce*distance
1/2 m vfinal^2=changeKE
To solve this problem, we need to break it down into three parts:
A: Finding the net force on the suitcase on the ramp.
B: Calculating the change in kinetic energy.
C: Determining the final velocity of the suitcase at the bottom of the ramp.
Let's solve each part step-by-step:
A: Finding the Net force on the Suitcase on the Ramp
1. First, we need to find the force due to gravity acting on the suitcase parallel to the ramp. This force can be calculated using the formula:
Fg_parallel = mass * acceleration due to gravity * sin(theta)
where mass = 10.0 kg, acceleration due to gravity = 9.8 m/s^2, and theta = 30.0 degrees (converted into radians).
Plugging in the values, we get:
Fg_parallel = 10.0 kg * 9.8 m/s^2 * sin(30.0 degrees)
Fg_parallel ≈ 49 N
2. Next, we need to find the force due to friction acting on the suitcase. This force can be calculated using the formula:
F_friction = coefficient of friction * normal force
To find the normal force, we need to find the force due to gravity acting perpendicular to the ramp:
Fg_perpendicular = mass * acceleration due to gravity * cos(theta)
Fg_perpendicular = 10.0 kg * 9.8 m/s^2 * cos(30.0 degrees)
Fg_perpendicular ≈ 84.85 N
Therefore, the normal force is equal to the force due to gravity perpendicular to the ramp:
normal force = Fg_perpendicular
Now we can calculate the force due to friction:
F_friction = 0.35 * 84.85 N
F_friction ≈ 29.69 N
3. Finally, we can find the net force on the suitcase on the ramp. Since the suitcase is moving downhill, the friction force opposes the force due to gravity parallel to the ramp. Therefore, the net force can be calculated as:
Net force = Fg_parallel - F_friction
Net force = 49 N - 29.69 N
Net force ≈ 19.31 N
So, the net force applied to the suitcase while it's on the ramp is approximately 19.31 N.
B: Calculating the Change in Kinetic Energy
To calculate the change in kinetic energy, we need the initial and final velocities. Since the suitcase starts from rest, the initial velocity is 0 m/s.
1. We can find the final velocity of the suitcase at the bottom of the ramp using the equation of motion:
vf^2 = vi^2 + 2 * acceleration * distance
Here, the acceleration is the component of acceleration due to gravity parallel to the ramp, which is given by:
acceleration = acceleration due to gravity * sin(theta)
vf^2 = 0^2 + 2 * 9.8 m/s^2 * sin(30.0 degrees) * 3.00 m
vf ≈ 5.39 m/s
2. The change in kinetic energy can be calculated using:
ΔKE = 0.5 * mass * (vf^2 - vi^2)
Since the initial velocity is 0, the equation simplifies to:
ΔKE = 0.5 * mass * vf^2
ΔKE = 0.5 * 10.0 kg * (5.39 m/s)^2
ΔKE ≈ 145 J
Therefore, the change in kinetic energy is approximately 145 J.
C: Determining the Final Velocity at the Bottom of the Ramp
The final velocity at the bottom of the ramp is already calculated above:
vf ≈ 5.39 m/s
So, the suitcase is traveling at approximately 5.39 m/s when it reaches the bottom of the ramp.
To answer these questions, we need to use some principles from physics. Let's break down the problem into smaller steps and find the solutions step by step.
Step 1: Calculate the gravitational force:
The gravitational force acting on the suitcase can be calculated using the formula: F_gravity = m * g, where m is the mass of the suitcase (10.0 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²). Thus, F_gravity = 10.0 kg * 9.8 m/s² = 98.0 N.
Step 2: Resolve the gravitational force into components:
Since the ramp is inclined at an angle of 30.0 degrees, we need to resolve the gravitational force into two components: one perpendicular to the ramp and the other parallel to the ramp. The component perpendicular to the ramp is F_perpendicular = F_gravity * cos(30.0°), and the component parallel to the ramp is F_parallel = F_gravity * sin(30.0°).
Step 3: Calculate the normal force:
The normal force is the force exerted by the ramp on the suitcase perpendicular to the surface. It cancels out the perpendicular component of the gravitational force. The normal force is given by N = F_perpendicular.
Step 4: Calculate the frictional force:
The frictional force acting on the suitcase is given by the formula: F_friction = μ * N, where μ is the coefficient of friction (0.35) and N is the normal force. Thus, F_friction = 0.35 * N.
Step 5: Calculate the net force on the suitcase:
The net force on the suitcase is the difference between the parallel component of the gravitational force and the frictional force. So, the net force is F_net = F_parallel - F_friction.
Step 6: Calculate the change in kinetic energy:
The change in kinetic energy of the suitcase can be calculated using the formula: ΔKE = KE_final - KE_initial, where KE_initial is 0 (starting from rest) and KE_final can be calculated using the formula: KE_final = (1/2) * m * v², where m is the mass of the suitcase and v is its velocity at the bottom of the ramp.
Step 7: Calculate the velocity at the bottom of the ramp:
The final velocity of the suitcase at the bottom of the ramp can be calculated using the formula: v = √(2 * g * h), where g is the acceleration due to gravity and h is the height of the ramp (3.00 m) * sin(30.0°).
Now, let's calculate the answers to each of the questions:
A: Net force on the suitcase:
- Calculate F_perpendicular = F_gravity * cos(30.0°).
- Calculate N = F_perpendicular.
- Calculate F_friction = μ * N.
- Calculate F_net = F_parallel - F_friction.
B: Change in kinetic energy:
- Calculate KE_final using KE_final = (1/2) * m * v².
C: Velocity at the bottom of the ramp:
- Calculate h = 3.00 m * sin(30.0°).
- Calculate v = √(2 * g * h).
Following these steps, we can find the solution to each question.