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Answer :
Jasmine must apply a force of approximately 27.7485 newtons, directed along the incline, to momentarily accelerate the cart at 0.765 m/s².
To find the force Jasmine must apply to the cart to momentarily accelerate it up the ramp, we can use the following steps:
Step 1: Calculate the gravitational force acting on the cart.
The gravitational force on an object can be calculated using the formula:
\(F_{gravity} = m \cdot g\)
Where:
- \(F_{gravity}\) is the gravitational force (in newtons).
- \(m\) is the mass of the cart (36.3 kg).
- \(g\) is the acceleration due to gravity (approximately 9.81 m/s²).
\(F_{gravity} = 36.3 kg \cdot 9.81 m/s^2\)
\(F_{gravity} ≈ 356.583 N\)
Step 2: Resolve the gravitational force into components.
Since the ramp is at an angle of 14.3 degrees to the horizontal, we need to find the component of the gravitational force that acts along the incline. This component is given by:
\(F_{gravity\_along\_ramp} = F_{gravity} \cdot \sin(\theta)\)
Where:
- \(F_{gravity\_along\_ramp}\) is the component of the gravitational force along the incline.
- \(F_{gravity}\) is the gravitational force we calculated in step 1.
- \(\theta\) is the angle of the ramp (14.3 degrees).
First, we need to convert the angle from degrees to radians because trigonometric functions typically use radians:
\(14.3^\circ \text{ to radians} = \frac{14.3}{180} \cdot \pi ≈ 0.25 \text{ radians}\)
Now we can calculate the component along the incline:
\(F_{gravity\_along\_ramp} = 356.583 N \cdot \sin(0.25 \text{ radians})\)
\(F_{gravity\_along\_ramp} ≈ 356.583 N \cdot 0.2474\)
\(F_{gravity\_along\_ramp} ≈ 88.366 N\)
Step 3: Calculate the force Jasmine must apply.
To accelerate the cart up the ramp at 0.765 m/s², we can use Newton's second law of motion:
\(F_{net\_along\_ramp} = m \cdot a\)
Where:
- \(F_{net\_along\_ramp}\) is the net force along the incline that Jasmine needs to apply.
- \(m\) is the mass of the cart (36.3 kg).
- \(a\) is the acceleration (0.765 m/s²).
\(F_{net\_along\_ramp} = 36.3 kg \cdot 0.765 m/s^2\)
\(F_{net\_along\_ramp} ≈ 27.7485 N\)
So, The answer is approximately 27.7485 newtons.
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Final answer:
Jasmine must apply approximately 90.7 newtons of force parallel to the ramp incline to accelerate the cart at 0.765 m/s², considering the forces due to both gravity and the required acceleration.
Explanation:
To determine the force Jasmine must apply to the cart to accelerate it up the ramp, we need to consider the components of force parallel to the ramp's incline. The force due to gravity acting down the ramp (the component of the weight of the cart parallel to the ramp) must be overcome by Jasmine's push. We also need to account for the force required to accelerate the cart.
The force of gravity parallel to the incline is calculated by mg sin(θ), where m is the mass of the cart and θ is the angle of the incline. To find the total force Jasmine must exert, we use Newton's second law, F = ma, where F is the total force exerted parallel to the ramp, a is the acceleration, and m is the mass of the cart. Hence, F = mg sin(θ) + ma.
Plugging in the provided values:
F = (36.3 kg)(9.8 m/s²)(sin(14.3°)) + (36.3 kg)(0.765 m/s²) = 90.7 N (approximately).
Thus, Jasmine must apply a force of about 90.7 newtons parallel to the incline to accelerate the cart at the specified rate.
