A box of textbooks of mass 24.8 rests on a loading ramp that makes an angle with the horizontal. The coefficient of kinetic friction is 0.25 and the coefficient of static friction is 0.35.
At this angle, how fast will the box be moving after it has slid a distance 4.9 along the loading ramp?
To determine the speed at which the box will be moving after sliding a distance along the loading ramp, we need to consider the forces acting on the box.
1. Determine the Normal Force (N): The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, it counters the weight of the box. The normal force can be calculated using the formula:
N = mg
where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s²).
N = 24.8 kg × 9.8 m/s²
N = 243.04 N
2. Determine the Force of Gravity (Fg): The force of gravity acting on the box can be calculated using the formula:
Fg = mg
where m is the mass of the box and g is the acceleration due to gravity.
Fg = 24.8 kg × 9.8 m/s²
Fg = 243.04 N
3. Determine the Force of Friction (Ff): The force of friction can be divided into two categories: static friction and kinetic friction.
The static friction force (Fs) will need to be overcome to set the box in motion. It can be calculated using the formula:
Fs = μs × N
where μs is the coefficient of static friction and N is the normal force.
Fs = 0.35 × 243.04 N
Fs = 85.06 N
Once the box starts moving, the kinetic friction force (Fk) will act on the box and can be calculated using the formula:
Fk = μk × N
where μk is the coefficient of kinetic friction and N is the normal force.
Fk = 0.25 × 243.04 N
Fk = 60.76 N
4. Determine the Net Force (Fnet): The net force is the vector sum of all the forces acting on the box. In this case, since the box is sliding along the ramp, the net force can be calculated using the formula:
Fnet = Fg × sin(θ) - Fk
where θ is the angle of the ramp.
Fnet = 243.04 N × sin(θ) - 60.76 N
5. Determine the Acceleration (a): The acceleration of the box can be calculated using Newton's second law of motion:
Fnet = ma
where m is the mass of the box and a is the acceleration.
Fnet = 243.04 N × sin(θ) - 60.76 N
243.04 N × sin(θ) - 60.76 N = 24.8 kg × a
Solve this equation for acceleration (a).
6. Determine the Final Velocity (vf): To find the final velocity of the box after sliding a distance of 4.9 m along the ramp, we can use the kinematic equation:
vf² = vi² + 2ad
where vi is the initial velocity (0 m/s), a is the acceleration, and d is the distance.
Solving this equation for vf gives:
vf = sqrt(2ad)
where vf is the final velocity.
vf = sqrt( 2 × a × 4.9 m)
By applying these steps, you can calculate the final velocity of the box.